Is Nathan Drake really Francis Drake's descendant?

From the Wikia,

In honor of their mother's belief that Francis Drake had living heirs, they adopt the "Drake" surname.

This does more than rule out his relation by name alone. This insinuates that Nathan and Sam had no reason to believe they themselves were the descendants of Sir Francis Drake. They simply believed that there were undocumented descendants and adopted the name in honor of that belief.

That said, the writers never explicitly said one way or another whether Nathan was actually a descendant or not.

In short, the answer is a definite maybe (but probably not).