Split Database tests in 'core' ones and 'driver specific' ones

Problem/Motivation

Follow up to #3129043: Move core database drivers to modules of their own and to #3230714-11: ConnectionUnitTest should be skipped for any database not psql or mysql:

Once #3129043: Move core database drivers to modules of their own is complete, it would be good to split the Database tests in 'core' ones and 'driver specific' ones, with the 'driver specifc' ones running dependent on the db driver in use by the SUT.

Steps to reproduce

Proposed resolution

Remaining tasks

User interface changes

API changes

Data model changes

Release notes snippet