Flag errors from psql when restoring from backups

When a database restore from PostgreSQL finishes, it's easy to miss
important errors that cause significant issues down the road. For
example, in https://gitlab.com/gitlab-org/gitlab/-/issues/36405, the
primary key constraint in `application_settings` was not able to be
created due to duplicate keys present.

With this change, we now:

1. Track all messages from stderr

2. Filter out messages with `does not exist`. These are present because
`pg_dump` is run with the `--clean` argument to issue DROP statements,
but `--if-exists` will filter this out automatically
(https://gitlab.com/gitlab-org/gitlab/-/merge_requests/40792).

3. If there are any error messages, play them back to the user with
a warning that these errors may impact GitLab.

4. Prompt the user to continue.