[.github/shared] Align eslint with .github

[mikeharder]

• Forgot to update .github/shared in last two changes
  • [.github] Add typescript-eslint #36181
  • Replace deprecated tseslint.config() with defineConfig() #37926

[github-actions]

Next Steps to Merge

Next steps that must be taken to merge this PR:

• ❌ The required check named Protected Files has failed. Refer to the check in the PR's 'Checks' tab for details on how to fix it and consult the aka.ms/ci-fix guide

Comment generated by summarize-checks workflow run.