Remove all `TEST_ID`s and the `check_test_ids.sh`

## Brief feature description

The `TEST_ID`s are no longer required for requirements tracing and can therefore be removed. The CI check script can be removed as well.

This would clean up the tests a bit.

Search the repository for `TEST_ID` and `check_test_ids.sh`, then inspect the CI configuration for references to the script. Done means the test IDs and script are removed without remaining references, and the test and CI checks still pass.

Written by the indexing model from the issue text.