For example I know that people use databases for frame data for various games but I would like more insight into this

Curious what kind of more insight you'd be looking for. I don't know how much experience you have with database stuff, but I don't think there'd be a whole lot of information to give you. They're basically just spreadsheets with fancy search functions.

That said, there's this app that's super useful that has frame data for various games (it has a web page as well, but I think it's mostly known as a mobile app), and it stores all the frame data in JSON files.

It's an open source project (link), and you can see where all the frame data is stored here.

Not really a database like you're looking for, but figured I'd shout it out since you explicitly asked about frame data.

Or if commentators or event organizers use databases to gather information on player stats that commentators use during tournaments.

Generally tournaments are hosted on smash.gg and all that information is kept on that site. They have an API for developers or whoever to grab that information, however. Again, not really a database like you're looking for. Though the information could potentially be stored in a database on smash.gg's side, it's not what the public is seeing.