Fix/fab auth manager race condition

[Pyasma]

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Was generative AI tooling used to co-author this PR?

This PR solves The issue of Race Condition Issue closes #63279
#63833

• Yes (please specify the tool below)

Claude 4.6

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• For fundamental code changes, an Airflow Improvement Proposal (AIP) is needed.
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[github-actions]

Backport failed to create: v3-1-test. View the failure log Run details

Note: As of Merging PRs targeted for Airflow 3.X
the committer who merges the PR is responsible for backporting the PRs that are bug fixes (generally speaking) to the maintenance branches.

In matter of doubt please ask in #release-management Slack channel.

Status	Branch	Result
❌	v3-1-test

You can attempt to backport this manually by running:

cherry_picker e7e188d v3-1-test

This should apply the commit to the v3-1-test branch and leave the commit in conflict state marking
the files that need manual conflict resolution.

After you have resolved the conflicts, you can continue the backport process by running:

cherry_picker --continue

If you don't have cherry-picker installed, see the installation guide.

[Pyasma]

@vincbeck thanks for merging this PR