Not really. I mean, it can happens, but it's much rarer than ship wars that share characters. I think the issue with Poe/Finn versus Kylo/Hux is some people not liking one pairing and wanting to see less of it on certain sites.

I've seen people get angry about Rey/Kylo and Hux/Kylo, but it's still not what I would consider a ship war since most of the anger is coming from people who don't ship either of the pairings.

The only thing I would consider even close to a ship war in fandom is the arguments about Poe/Finn versus Finn/Rey, but those tend to be about who's more "progressive" with their ship. I find the whole thing a bit mind boggling because either of them being canon would be a great step forward for gay couples or interracial couples. I know people complain about not wanting another het ship, but there are still a lot of people who are upset at the thought of a white woman being with a black man and there aren't many positive representations of interracial couples in major blockbusters, so it would still be "progressive" even though it's still a het ship - and I say this as someone who ships every variation of Poe/Finn/Rey.