Do you think a patronus and an animagus need to be, or are likely to be, the same animal for a person?

Likely, yes; necessary, no. We know that patronuses can take forms it's not possible to become as an animagus, because Tonks's patronus becomes a werewolf after falling in love with Lupin. (I guess we don't rightly know you can't choose a werewolf as your animagus form, but the idea feels janky, like it's not reeeeeally what the magic system was conceived to do.)

My general philosophy of Patronuses gets explored a lot in an upcoming chapter, but the TLDR is I think they're a token of something that makes you happy. Harry's is a stag because his connection to his father brings him joy; Tonks, a representation of Remus; Snape, a representation of Lily; etc. That's also why patronuses can change when you seek joy/meaning from new places. Nowhere in the books do we see the kind of "spirit animal" reading that crops up in personality quizzes about it. Yeah, sometimes the animals happen to fit their characters, but why do McGonagall and Umbridge — two very different characters — have the same patronus? Wouldn't it make more sense if they were just two dames who like cats?

In contrast, your animagus form is said (extracanonically, so Doesn't Count, if you ask me, but whatever) to be involuntary and based on your personality. (This is part of the reason I don't think you can use animagus abilities to become a werewolf, because what would that even mean about the person doing it?) The animagus is supposed to function as a "spirit animal" type of thing, because it's technically still you.

So if your patronus and animagus form happened to match, I'd read it as a good indication that the things bringing you joy are closely related to things you like about yourself. It would be an indication of strong self-esteem and satisfaction. James Potter being one such character, and someone who's established in canon to think pretty highly of himself. Minerva McGonagall is a god damn delight and she knows it. Interestingly, Lupin's patronus is a wolf, but not a werewolf, which could mean any number of things — for one, the word "Lupin" being a cognate for wolf in Latin means that his patronus could reference his family or parents, or just his own identity as a person outside of his lycanthropy.