chore: refactor shared code between services

[ohmayr]

Thank you for opening a Pull Request! Before submitting your PR, there are a few things you can do to make sure it goes smoothly:

• Make sure to open an issue as a bug/issue before writing your code! That way we can discuss the change, evaluate designs, and agree on the general idea
• Ensure the tests and linter pass
• Code coverage does not decrease (if any source code was changed)
• Appropriate docs were updated (if necessary)

Fixes #<issue_number_goes_here> 🦕

[ohmayr]

Notice: Migration in Progress

This repository is being consolidated into the google-cloud-python mono-repo. We are closing this PR here to prevent "split-brain" development during the transition.

What happens next?

• Your work has been captured in our migration snapshot.

• We are evaluating all active PRs for manual porting into the new location: packages/gapic-generator-python/.

• You can track the porting status and the technical migration plan here: [gapic-generator] Port active PRs and logic from legacy gapic-generator-python repository google-cloud-python#16182