I can't be the only who how thinks this is most likely the answer. Like I said, none of it adds up. Considering when they both died, Credence is too young to be the son of either Dumbledore parent. Also, if Credence was truly a Dumbledore that his siblings knew of (which Grindelwald seems to imply Albus does), Grindelwald would have known who he was when they met in Fantastic Beasts.
He and Albus were far too close growing up for that little secret not to have been shared before. Another clue to Grindelwald lying is that he tells Credence that he's giving him the name that should have been his birthright, implying that he is naming him and it isn't his actual birth name. As for the phoenix appearing to him "in his time of need," we've seen a phoenix help out people who aren't Dumbledores — Harry Potter for instance — so I'm taking that with a grain of salt. Maybe Credence is that good with animals that a wild phoenix took a liking to him.
But why would Grindelwald lie about Credence being Albus's brother? It's the perfect way to turn him against the other man. By taking him in when his "family" seems to work against him, Grindelwald ensures that Credence will follow his plan and work for him. He gets the power of an Obsurus and a tool to destroy Albus, since he can't personally fight the other wizard.