It doesn’t.
Both DX and K2A-O open a local keepass file.
They are capable of reloading the file when it is changed, and can be set to immediately write out changes to the file.
Then you take whichever file sync tool you like and sync it with all other devices using it. As long as the sync tool can sync files in your internal storage, it will work.
I use syncthing, with a dedicated keepass folder containing only the database file. Then I simply add all my devices to the share and it’ll sync any changes to all other devices. I also have version history enabled for the share.
The offline version is on izzyondroid.
The design is worse, yes.
I don’t think it matters much because most of the time you only see the autofill thing, not the app.
When you do go to the app, it is to select between multiple credentials, which is still a split second action.
On mobile I have my 2fa in a different more convenient app (aegis), though k2a does allow to copy 2fa codes