can a club become rivals with another club during a game?
like over a number of years and stuff?
Yes it can, for example i became rival of Nacional da Madeira with Benfica.
Intresting thing to know. I never thought that it could happen.
On a side note, do you know if it's possible for a team to stop being a rival? For instance West Ham got relegated from the premier league in my first season and I haven't played them since and they're still a rival.