Eh. It sort of... varies, depending on who's writing and what angle you're looking from and what mood Nasu's in.
The original FSN, for example, treats the decision to have Sasaki Kojiro be a fictional, invented character (implicitly positioning every other mythological figure in the story as an actually real person who genuinely existed, which they pretty much have to be when they directly contradict the popular image of that character) as something important and meaningful. The resulting Servant is not Sasaki Kojiro, he is just Some Dude who never got famous enough to make it to the Throne of Heroes, but had the insane skills needed to approximate the role of Sasaki Kojiro. He's a ghost stuffed into a Sasaki Kojiro costume, and as a result he doesn't even have a "proper" Noble Phantasm (inasmuch as it matters).
Fate/Extra returns to play with this idea using Robin Hood (there were a bunch of "real" Robin Hoods, this guy was just one of the people to adopt the title, and doesn't think he even did a very good job), Vlad Tepes (a historical man with an invented vampiric legend, driven mad by the metaphysical influence of the latter, which manifests as a Skill in its own right, Innocent Monster), and Nursery Rhyme (she seems like a Servant version of Lewis Carrol's fictional Alice, but actually there's weirdness going on and she's not even really a Servant so much as a dying child's magical tulpa).
Apocrypha plays with this idea even more explicitly by returning to Vlad Tepes, who is a historical figure who is not Dracula, because Dracula was a fictional character and was not real, even if everyone thinks he's Dracula. Notably, when Vlad's forced to use the Noble Phantasm which transforms him into Dracula, he is not a Dead Apostle - which is what a "real" historical vampire would be in this setting. He's an expression of the entirely fictional character Dracula, with all the powers and weaknesses that entails. Of course, Apocrypha also has Frankenstein's Monster just... show up, apparently as a real person and actual Heroic Spirit, but that's far from the worst writing in that story.
Fate/Zero doesn't really touch on it except vaguely through Bluebeard, who is the actual historical figure Gilles de Rais, not the invented fairy tale character Bluebeard, despite bearing that title and being one of the character's inspirations.
A lot of fanfic writers took Sasaki Kojiro's original scenario as license to do silly shit like summon Harry Potter or the Hulk, but setting aside the tonal issues, that's not really how Sasaki worked. He wasn't a fictional character summoned as a Heroic Spirit to be a Servant. He was a real dude, summoned to fill the shoes of a Heroic Spirit who could have existed, but unfortunately wasn't real. A real ghost wearing an empty legend like an ill-fitting suit.
FGO then skids wildly all over the map, because a) we want to do a bunch of obviously fictional Servants but we don't have the time, space, or motivation to run through existential musings for all of them, b) there's no time, we need five Singularity scripts out the door yesterday and we don't care if it shows, c) Nasu changed his mind, pick between one and three options.
Then Sherlock Holmes shows up as a major character and Heroic Spirit, so he's clearly a real person despite being famously fictional in-setting, Mash has read the books written by an author about this fictional character. Except no, he's actually a crystallisation of the whole detective genre and the idea of mystery revealers, presumably built around the spirit of an anonymous detective or Joseph Bell. Except no, here's a spin-off story where he interacts with real actual historical characters, so he's a real person again. Except no, his profile is acting all coy about whether he's real, and here's Moriarty teasing the idea that neither of them are real people, and Holmes gets uncomfortable about addressing it. And in fact, Moriarty's whole EoR chapter revolves around the conceit of "Phantom Spirits" who are too vague, obscure, or transparently fictional to become true Heroic Spirits, but can be fused with each other or used to alloy the legends of actual Heroic Spirits.
The Count of Monte Cristo is a Heroic Spirit and definitely a real person, because he killed a significant character (Roa) in an actual historical battle that really happened in the history of the setting. The author of the fictional work, the Count of Monte Cristo, which he wrote as fiction, is also definitely a real person who exists in the setting, because a) he wrote the damn book, b) he also gets summoned as a Heroic Spirit. So you know what, it's all whatever, just stare vaguely into the middle distance and mutter something about "world layers" and "consolidation of mystery" and "something something Second Magic something".
Anyway, going by the examples presented, the most consistent things that would happen if you summoned J.R.R. Tolkien's Sauron would be a) Sauron was actually a real person in the setting, I guess, don't think too hard about it, b) you summoned Andvari, the shapeshifting dwarf from Norse mythology who cursed the magical ring Andvaranaut at the centre of Der Ring des Nibelungen, and you're just calling him Sauron because you misunderstood and the Holy Grail thinks it's funny, c) you summoned the ghost of some random historical ringsmith, possibly magical, who went unrecorded in history, and he got stuffed into a really flimsy Sauron costume because you're an idiot, d) you summoned some kind of demon that is using the accumulated concept of Sauron, and by extension a jumble of more firmly established ideas around Lucifer, Gyges, the tempting devil, etc, as an anchor and vessel for existing in this world as a Servant.
I'm placing money on d), for what it's worth, because to my recollection Sauron wasn't ever really much into the Lovecraftian SAN damage, so much as... making very tempting offers and letting people destroy themselves.