I think this is better asked as, "Under what circumstances do historians engage in debate about whether a particular historical figure actually existed?"

For the most part, we don't debate the actual existence of any named, known individual that appears in archival records of any kind created in the last two centuries. Only very rarely do we decide that an individual described as a real, literal human being simply did not exist--usually because they were described by a single author who was later on revealed to be creating a hoax. The Klan member Asa Earl Carter created a fictional persona of "Forrest Carter" in writing The Education of Little Tree, a book that was originally sold and read as if it were an autobiography, until the historian Dan T. Carter revealed the hoax in a 1991 essay.

Sometimes historians find that a modern person who really existed nevertheless did not live the actual life commonly attributed to them--Tim Couzens' biographical study of Trader Horn demonstrated that many of his claims about his life when he became famous in his old age were exaggerated or made-up. Those kinds of arguments can go on for a long time (or never really be resolved)--say, in the case of the author Laurens van der Post, who some scholars believed made up a good portion of his autobiographical writings and others think was guilty only of occasional small exaggerations. No one doubts that such people actually existed, however--you have to get back to the 17th-18th Century before significant numbers of named individuals begin to be the subject of controversies about whether they actually existed or not.

Even in older period, most of the time if you have documents written by a specific individual with that specific individual's name on the documents and you have a strong verification of the provenance of those documents (e.g., you have no reason to think they are faked), you tend to assume that the author of those materials really existed as a real, specific person. If I'm reading letters from 1685 from one person to another person, I tend to assume that both of them were real people, even if I have nothing besides those letters verifying their existence. Once you go back before the 19th Century, in most places in the world, you're only very rarely going to come across governmental or institutional entities that in some way or another verify the existence of specific individuals via record-keeping (say, of taxation, of birth or death, of marriage). But if you've got a named person creating a document, or even a named person being described directly by a writer as someone they've really met and interacted with on a regular basis, you don't look for reasons to doubt that those people are real, even if you may doubt many aspects of their lives as described in those materials.

I think there are two main circumstances in which a historian has to grapple with the question of the literal existence of a named individual in the more distant past. Generally, the further back you go, the less chance you have of ever really resolving your doubts in a truly final or satisfying way. The first circumstance is when you have some more recently created record of a genealogy or lineage, often because a historian or official asked people to describe their kinship history. You often find that somewhere between five to eight generations back, these get very hazy. Even if you're talking to people who share some part of that kinship, you'll hear different names for what seem like the same people. Some lineages will put a particular person in one generation, some will put him or her in another. If it becomes important to connect these stories to a specific individual about whom there is some other kind of evidence--say a traveller's report about people in this area, etc.--then you sometimes have to decide whether a particular named ancestor is a single literal person or is instead some kind of combined or conflated imagined person who refers to multiple ancestors who are poorly remembered because of how long ago they lived.

The second reason, far more commonly a part of debates among historians, is when we know about a past individual largely through legends, folklore, or historical accounts written decades or even centuries after that person allegedly existed. There are many, many examples of such individuals, all of whom have provoked lengthy scholarly and non-scholarly discussion about whether they really existed, and if they did exist, whether they at all resemble the person described in those accounts. Robin Hood, King Arthur, Jesus of Nazareth, the early kings of Rome like Lucius Tarquinius Priscus, Laozi, Socrates, etc.: there are a huge number of such figures where there is at least some reasonable debate about their literal existence.

In general, those discussions come down in favor of someone having been a real individual when there is some roughly contemporaneous source that refers to them by name and associates the name with a substantial number of tangible, specific biographical details, when there are multiple sources that are near to the time that person existed which provide the same or similar biographical details independently of each other, and when such sources seem largely literal or matter-of-fact in their tone and aren't visibly trying to build up the individual as a legendary or extraordinary figure. So we have multiple mentions of Socrates by Athenian contemporaries, some of them substantial--so no doubt that he was a real person. (It's true that Plato's version of "Socrates" is something of a device for Plato's own philosophy, but that doesn't make Socrates an imaginary person.)

On the other hand, discussions trend towards thinking that no such person existed, or that any real individual was so different from the version we know about that linking the legend and the person is a dubious exercise, when we have absolutely no records or accounts of that person from anywhere near the time they supposedly lived, when the accounts of their lives have blatantly inaccurate or impossible elements in them and seem to promote the legendary deeds or character of the individual, and when we don't even have a plausible chain of oral transmission between the time of the individual and the first written or recorded version of their lives. King Arthur and Robin Hood are the classic examples. Sure, maybe there's a ruler somewhere in British history who inspired some of the later folklore about Arthur; maybe there was an actual bandit who was the original Robin Hood. But just as likely, the Arthur stories are a synthesis of numerous mythological and folkloric narratives that refer to no real person or combine elements of many actual individuals; Robin Hood might well have just been a generic title that medieval English bandits used, or that was attributed to them and trying to document the 'real' Robin Hood is thus as futile as trying to prove that Batman was a real person in the 20th Century.