Some genres have more commonality than others. Blues songs often have a similar 12-bar structure, for instance. Not all Blues songs, of course, but there are some songs you can listen to and quickly identify it as Blues based on the structure. This structure exists to improvise on top of. Jazz has a lot of improvisation built into it too.

I also think genres in the past were based on which radio stations they were played on, back when radio was the main way to hear new music. “Pop” music simply meant “popular”, was meant to be more broadly acessible, and was played on Top 40 stations. Whatever counts as “pop” changes with the times. Now, while the distinctions still exist, I don’t think most young people get their music from the radio anymore, so the genres ar not to rigidly defined.

What I think it comes down to is that bands identify themselves based on whatever they listened to, and what influenced them. So the best way to know what genre a band plays is to ask them.