That seems to be the reason, thank you Adrian.
At first connecting from the other pod in the AKS cluster does not work:
$ PGPASSWORD=timeshift_pass psql --host=timeshiftservice --port=5432
--dbname=timeshift_database --username=timeshift_user
psql: error: connection to server at "timeshiftservice" (10.0.120.194),
port 5432 failed: FATAL: password authentication failed for user "t
imeshift_user"
But once I connect to my PostgreSQL pod (or is it called "container"? I am
always confused by that) and run locally
$ psql -U postgres
psql (17.4)
Type "help" for help.
timeshift_database=# ALTER ROLE timeshift_user PASSWORD 'timeshift_pass';
ALTER ROLE
Then the connection from the other pod works immediately