1. If a service supports sqlite, I often will use that option. It provides everything a self-hoster needs from a DB with basically no operational overhead.
2. If I do need a proper RDBMS (because the software I'm using doesn't support sqlite), I'm going to use...
   1. A single Postgres container.
   2. Configured with multiple logical "databases" (the container for schemas and tables), one DB for each app connecting.

I do this because I'm always memory constrained and the rdbms is generally the most memory-hungry part of any software stack. By sharing one db-process across all the apps that need it I get the most out of my db cache memory, etc. And by using multiple logical db's, I get good separation between my apps, and they're straightforward to migrate to a truly isolated physical DB if needed... but that's never been needed.