Katsuki’s type of music would be anything that is easy listening. Soft rock, anything quiet. Surprisingly, listening to that kind of music helps him feel relaxed and calm.
Izuku likes to listen to pop, anything up-beat and happy. (I’d say he listens to a lot of j-pop/k-pop).
Katsuki hates the kind of music Izuku listens to, but Izuku really likes Katsuki’s. Whenever Izuku listens to his music, his grouchy boyfriend wishes he’d just turn it off, or he actually does turn it off. It results in Izuku pouting, but then he’ll turn on the type of music Katsuki likes, so that he stops complaining.