Yes and no. If we find some kind of solid record of their name or accomplishments, especially by a third party, that's usually taken as pretty solid evidence. Treaty documents are really great for this, as are inscriptions on buildings or monuments. Letters or other writings by the figure can also be really good, but tricky. For example, there isn't serious doubt about the existence of Plato (although we do question the authenticity of a few of the writings attributed to him), but your designation of Homer as real comes with a lot of asterisks (I won't go into them because that's not my field).

However, this all depends on the source, and in my field (biblical studies/early Christianity), two really important standards are doubtfulness and utility. We don't have any reason to doubt the existence of John the Baptist. His treatment in the Gospels is clearly the result of some embarrassment--he's shown as simultaneously endorsing and competing with Jesus. In some ways it would be easier just to leave him out, but clearly the Gospel authors don't think they can. His treatment in Josephus is more dry, and while Josephus can't always be trusted for his interpretations of events, he doesn't get anything out of inventing someone like John.

(Embarrassment, by the way, is a great standard for things like this--if someone looks like they're trying really hard to justify something or force it to fit with their own agenda, it's a good signal that the thing might be important outside of their agenda.)

Now let's look at, say, Blind Bartimaeus from Mark 10. He plays a narrative role in the plot of the Gospel, doesn't have a real name ("son of shame" or something like that), and there's no reference to him in any sources that don't depend on Mark. That's not to say he for sure wasn't real, but there are a lot of reasons to doubt.

Contrast all that to Jesus. We don't have any uninterested third party references to Jesus (the reference in Josephus might not be genuine). We don't have anything he wrote. He wasn't rich or a politician, so we don't have his name carved on any buildings. Christianity obviously has a lot invested in his existence and the nature of it, so those accounts can't be taken as unbiased. However, there's very little debate within academic New Testament studies about the reality of Jesus. Part of that is because NT studies tends to be populated by believers, but it's also, on my end (I'm not a believer), down to the utility factor. It's not very useful to us to speculate on the reality or not of Jesus. Either way, we get Christianity. What we know about the first two or three decades of the movement comes entirely from Paul, only some of whose writings we have, and the Gospels. But Paul isn't very reliable for historical fact because that's not what he's trying to do and because his corpus is incomplete, and the Gospels date from 40 years after the crucifixion and later. So a whole lot of what we can say about the Jesus movement in the first century is speculation, anyway--if you add in "Also Jesus was invented by Peter" or something, you're just doing different speculation. It doesn't change anything in any concrete way, so it's not really a useful discussion to have from a historical standpoint (not getting into issues of belief here).

So the TL;DR is that it comes down to physical or other concrete evidence, including preserved original documents like treaties; literary evidence in non-fictional/non-mythological sources, including a person's own body of writing; doubtfulness and how the figure fits or doesn't into the agendas of the sources we have; and the usefulness of the question at all.

Note: Even after ALL THIS, establishing the reality of a person's existence is not the same thing as trusting that person's portrayal either of themselves or in secondary literature. It can be a really subtle distinction. There's no reason to doubt the existence of John the Baptist, but the sources we have for him are still using him to get at their agendas--we can't assume that we can know much about him for sure. There's no reason to doubt the existence of Paul, but homeboy has agendas coming out of the ears, many of which we don't understand that well, and the ways that he chooses to represent himself are very careful and thought out.