It's file-based. If you have a copy open in one location, open a copy in a second, save the second, then make changes in the first and save the first, you can override the second copy's changes.
That said, KeePass 2.x (which came out in 2010/2011, but some people still use the old version) has synchronization built-in, and on top of that, there are plugins to make it keep backups or update a secondary database. It's the KeePass clones that have their own implementation of the internals where you can run into problems.
IT does what it says on the tin; it's a password manager. However synchronisation is extra (I use sync to sync the file, and because I'm on iOS I have to manually import the file into the app).
The recommended iOS app (MiniKeepass) doesn't support Touch ID to unlock the database (a PITA with a long master password, which I already have to unlock my device), so I'm using keepass touch, which is a MiniKeepass fork, but with ads, and they've been ignoring my requests for source code (MiniKeepass is GPL).
Browser integration is a pain; you haveto install an extension to Keepass, and then to your browser, and set them up correctly. Steps are at [0]
It works, but is a lot more hassle (which as a result means I'm more likely to skip it and use an insecure password) than Lastpass was.
How does it not work? Confidentiality and data integrity are essential for password managers; they can't work most of the time in those regards.