MySQL 3.x was mostly just okay for simple website that didn't do much. (MyISAM is good for either extremely read heavy with few writes or vice versa, was very performant in those cases for the time)
MySQL 4.x added subquery support and InnoDB as a standard component, even if not the default.
MySQL 5.0 added cursors, triggers, stored procedures, views.
I think from about this point MySQL was fairly usable in place of some enterprise DB's depending on use case, and most of the essential features were there, even if not always the default.
MySQL started as a super easy to use and accessible DB, and slowly accreted enough enterprise level features to be able to compete in that field, but it's been in use successfully there for quite a while.
In comparison, my (super limited and possibly incorrect) understanding of Postgres's history is that they focused on stability and enterprise level features first, and then later added some things for more convenience. I think they aren't super different in capabilities now, at least for the core functionality most expect.
InnoDB has been the default storage engine since 5.5, released in late 2010, so for over a decade.