This is an opportunity to exercise some social nuance. To them, "MySQL" and "database" are equivalent, and to point out the difference is to make the social faux-pas of overspecification (somewhat of a trope for nerds like us).

By the time you're having this conversation with a BA, the DB is not going to be changed because they made some offhand reference to a different DB. They said something technically wrong, but in a way that doesn't have any meaningful effect on what they were trying to say: as such, to nitpick a detail like that may give the impression of trying to reinforce your superiority by pointing out tiny technical mistakes in their speech. So just let it go :)