tests: housekeeping, test_identity.py

housekeeping, the following is looked at and may have been done:
* fixed typos and standardized formatting
* renamed test cases to improve the clarity of what the test does
* improved docstring language, setup, steps and expected results
* synced code with the docstring order
* removed necessary configuration relevant to the test
* added pytest.mark.importance to test cases

noteable changes:
* auto_private_group tests reworked, adding two more test cases

Reviewed-by: Alejandro Lopez <[email protected]>
Reviewed-by: Jakub Vávra <[email protected]>
Reviewed-by: Madhuri Upadhye <[email protected]>