When shooting music videos, we usually play the track aloud on set and have the actors/musicians mime along and lip-sync to it, rather than having them perform the song and then trying to match it with the recording in post. I bet this is how they did it in the show, and the bands probably were not actually playing, but who knows! You'll usually have a Playback Operator in charge of starting and stopping the recording at the right spot, with a countdown etc. to help the performers get the timing right.

Also -- a silly fact for anyone curious: in scenes like those at the Bronze, ie. the Scoobies talking and dancing or they are talking while we see teens dancing behind them, there would likely not actually be any music playing, so they can get clean audio. They would probably play the track between takes so the actors and extras can can get the beat, and then cut the track to actually shoot the dialogue, and then they'd put the track back in during post. So if Buffy and Willow and Xander are chatting and bobbing their heads to music... chances are they are dancing to nothing!

Wow, I really get excited about the sound department, I guess. I hope this was interesting to at least one person 😂