is there some way to restrict the DB-users to just be able to modify and delete content they added to the db (from the backend side)?

MySQL allows you to control user access at the database, table and column level. See Chapter 4 of the manual for full details.

Depending on your application you might be able to use this to enable the kind of access control you want.

For example, you might create a table for each user, and only give the appropriate user insert/update permissions. You could then create a global view of all of those tables so everybody can read them.

Without some more details it's hard to tell. It might be easier to roll your own access control system.

With PostgreSQL, of course, you could add a triggers on the database side to roll your own access control system on the database side :-)