Is it obvious to others that he's fulfilling this role [as White Prophet]?
No--and yes.
In the Realm of the Elderlings canon, the religion of the White Prophet is unknown in most parts of the charted world. It began in Clerres, a small, remote part of the world with its own insular culture and language, which, aside from trade, does not seek to garner attention from other seafaring powers. In Clerres, individuals who are born with the Fool's peculiar pale white colouring (white hair, colourless eyes, a slightly inhuman appearance) are marked from birth as having White lineage, meaning that they possess the ability to prophesy the future through their dreams. Nearly all children born with this lineage have this ability, though only one per age is supposedly called towards the higher purpose of setting the course of history upon its correct course.
That individual is called the White Prophet, and this is who the Fool believes himself to be. While the White Prophet is able to see the future, they require the actions of a Catalyst--a person of special significance who “functions as an agent of change” for the White Prophet. For the Fool, this person is FitzChivalry Farseer, the bastard prince of the Six Duchies, as well as the Fool’s closest friend.
Clerres' isolation means that the White Prophet religion has little influence beyond its borders; given that the majority of the Fool's story takes place in the northern kingdom of the Six Duchies, far across the sea and worlds away from his home, almost no one he interacts with in canon knows of his role.
At its core, the religion is concerned with the fate of the world, and the ability of some individuals to see and predict the future through the prophetic power of their dreams. In an ideal world, this ability would be leveraged to promote justice and harmony for all sentient creatures; the Realm of the Elderlings, however, is not an ideal world, and in all aspects of life, decisions are politically motivated. When the Fool (who at this time was known only by his birth name, Beloved) began experiencing his prophetic dreams as a child and believed himself to be the White Prophet, he was informed by his teachers that a White Prophet for their age had already been identified--the Pale Woman. Also known as Illistore, she would later go on to be the one responsible for some of the worst atrocities in the history of the Six Duchies (known as the Red Ship Wars, and Forging, in the Farseer trilogy)--as well as the Fool's death at the conclusion of Fool’s Fate, the last book in the Tawny Man trilogy.
The final conflict between these two prophets--one true, one false, one bent on the creation of a better world, the other on destroying it--revolves around dragons, and their place in the Realm of the Elderlings. The Fool, following the path laid out before him in his dreams, strives towards the act of bringing them back to the skies, in a journey that spans nine books, six trilogies, and four different identities who are all ultimately just facets of himself. He dies for that ambition at the end of his journey--only to be revived by his Catalyst.
Why does the Fool feel cast adrift, and why is it necessary for him to take on a chameleonic role?
I’ll address the latter part first just because it feeds into the former! (NB: I think this section also addresses why the Fool avoids addressing personal inquiries about his past, as well as the role of his identities as means of expressing himself.)
The Fool’s roles. The extent of the abuse that the Fool experiences during his time in Clerres isn't explored or divulged by the narrative until well after the canon point I'm choosing to app him from and into Khu Ioduan, but knowing that it exists (and is very, very extensive), is useful to understand at least part of why the Fool chose to shroud his identity after he escaped from Clerres as a child. He runs away and takes on new roles and identities to protect himself--essentially hiding in plain sight.
1) When he flees his home, it's because the alternative is to deny what he believes to be the truth about his identity and his life’s calling, and to instead toe the party line by acknowledging Illistore as the White Prophet. It is also heavily implied that if he didn't flee, he will be tortured and killed so as to ensure Illistore has no competition for the title of White Prophet. If he runs away, it means that he will have a chance--albeit a very small one--of bringing about the future that he believes in. So he runs.
2) Part of running means building a new life for himself essentially from scratch, and as a vulnerable child alone in the world he quickly learns to play into people’s expectations of him as a means of blending in. His first ‘role’ after he leaves Clerres is where he acquires the moniker of the Fool, and later turns it into a name. He makes himself a jester in the court of Six Duchies monarch, King Shrewd, order to deflect serious attention away from his other-worldly appearance. Even though Clerres seems like it's worlds away, trade still happens, and with trade travel rumours and speculation. The servants of the White Prophet religion would be listening for news of any strange people with a peculiar appearance proclaiming himself to be a prophet of some kind.
He changes his identity multiple times throughout the course of his journey through the Realm of the Elderlings, and while it’s originally done as a means of blending in, he genuinely seems to delight in the theatre involved in each role that he plays. (Example: as the Fool, he is somewhat infamous within King Shrewd’s court for his relentless mockery of the nobility, finding ways to make them look foolish in public in the event he discovers that they are seeking to undermine King Shrewd in some way. As Lord Golden, he’s similarly infamous for his haughty and decadent behaviour, which gains him access to social circles and information he wouldn’t be able to touch otherwise.) I also speculate that through concealing his true name and becoming first the Fool, then later a woodcarver woman known as Amber, and then the fop Lord Golden, he uses these other identities as a means of expressing himself in a way that he feels he can’t otherwise, because to truly be himself openly--to be Beloved--is to invite attention from corners of the world he isn’t prepared to face yet: the Pale Woman and the Servants in Clerres.
(I also believe that his identity as Amber is his means of demonstrating his gender fluidity, since he takes extreme offence to Fitz accusing him of ‘pretending’ to be a woman when he discovers the Fool dressed as Amber at one point. When using the identity of Amber, the Fool uses she/her pronouns, and she does not shed these when in private.)
[1/2]
Is it obvious to others that he's fulfilling this role [as White Prophet]?
No--and yes.
In the Realm of the Elderlings canon, the religion of the White Prophet is unknown in most parts of the charted world. It began in Clerres, a small, remote part of the world with its own insular culture and language, which, aside from trade, does not seek to garner attention from other seafaring powers. In Clerres, individuals who are born with the Fool's peculiar pale white colouring (white hair, colourless eyes, a slightly inhuman appearance) are marked from birth as having White lineage, meaning that they possess the ability to prophesy the future through their dreams. Nearly all children born with this lineage have this ability, though only one per age is supposedly called towards the higher purpose of setting the course of history upon its correct course.
That individual is called the White Prophet, and this is who the Fool believes himself to be. While the White Prophet is able to see the future, they require the actions of a Catalyst--a person of special significance who “functions as an agent of change” for the White Prophet. For the Fool, this person is FitzChivalry Farseer, the bastard prince of the Six Duchies, as well as the Fool’s closest friend.
Clerres' isolation means that the White Prophet religion has little influence beyond its borders; given that the majority of the Fool's story takes place in the northern kingdom of the Six Duchies, far across the sea and worlds away from his home, almost no one he interacts with in canon knows of his role.
At its core, the religion is concerned with the fate of the world, and the ability of some individuals to see and predict the future through the prophetic power of their dreams. In an ideal world, this ability would be leveraged to promote justice and harmony for all sentient creatures; the Realm of the Elderlings, however, is not an ideal world, and in all aspects of life, decisions are politically motivated. When the Fool (who at this time was known only by his birth name, Beloved) began experiencing his prophetic dreams as a child and believed himself to be the White Prophet, he was informed by his teachers that a White Prophet for their age had already been identified--the Pale Woman. Also known as Illistore, she would later go on to be the one responsible for some of the worst atrocities in the history of the Six Duchies (known as the Red Ship Wars, and Forging, in the Farseer trilogy)--as well as the Fool's death at the conclusion of Fool’s Fate, the last book in the Tawny Man trilogy.
The final conflict between these two prophets--one true, one false, one bent on the creation of a better world, the other on destroying it--revolves around dragons, and their place in the Realm of the Elderlings. The Fool, following the path laid out before him in his dreams, strives towards the act of bringing them back to the skies, in a journey that spans nine books, six trilogies, and four different identities who are all ultimately just facets of himself. He dies for that ambition at the end of his journey--only to be revived by his Catalyst.
Why does the Fool feel cast adrift, and why is it necessary for him to take on a chameleonic role?
I’ll address the latter part first just because it feeds into the former! (NB: I think this section also addresses why the Fool avoids addressing personal inquiries about his past, as well as the role of his identities as means of expressing himself.)
The Fool’s roles.
The extent of the abuse that the Fool experiences during his time in Clerres isn't explored or divulged by the narrative until well after the canon point I'm choosing to app him from and into Khu Ioduan, but knowing that it exists (and is very, very extensive), is useful to understand at least part of why the Fool chose to shroud his identity after he escaped from Clerres as a child. He runs away and takes on new roles and identities to protect himself--essentially hiding in plain sight.
1) When he flees his home, it's because the alternative is to deny what he believes to be the truth about his identity and his life’s calling, and to instead toe the party line by acknowledging Illistore as the White Prophet. It is also heavily implied that if he didn't flee, he will be tortured and killed so as to ensure Illistore has no competition for the title of White Prophet. If he runs away, it means that he will have a chance--albeit a very small one--of bringing about the future that he believes in. So he runs.
2) Part of running means building a new life for himself essentially from scratch, and as a vulnerable child alone in the world he quickly learns to play into people’s expectations of him as a means of blending in. His first ‘role’ after he leaves Clerres is where he acquires the moniker of the Fool, and later turns it into a name. He makes himself a jester in the court of Six Duchies monarch, King Shrewd, order to deflect serious attention away from his other-worldly appearance. Even though Clerres seems like it's worlds away, trade still happens, and with trade travel rumours and speculation. The servants of the White Prophet religion would be listening for news of any strange people with a peculiar appearance proclaiming himself to be a prophet of some kind.
He changes his identity multiple times throughout the course of his journey through the Realm of the Elderlings, and while it’s originally done as a means of blending in, he genuinely seems to delight in the theatre involved in each role that he plays. (Example: as the Fool, he is somewhat infamous within King Shrewd’s court for his relentless mockery of the nobility, finding ways to make them look foolish in public in the event he discovers that they are seeking to undermine King Shrewd in some way. As Lord Golden, he’s similarly infamous for his haughty and decadent behaviour, which gains him access to social circles and information he wouldn’t be able to touch otherwise.) I also speculate that through concealing his true name and becoming first the Fool, then later a woodcarver woman known as Amber, and then the fop Lord Golden, he uses these other identities as a means of expressing himself in a way that he feels he can’t otherwise, because to truly be himself openly--to be Beloved--is to invite attention from corners of the world he isn’t prepared to face yet: the Pale Woman and the Servants in Clerres.
(I also believe that his identity as Amber is his means of demonstrating his gender fluidity, since he takes extreme offence to Fitz accusing him of ‘pretending’ to be a woman when he discovers the Fool dressed as Amber at one point. When using the identity of Amber, the Fool uses she/her pronouns, and she does not shed these when in private.)