refactor: auth store handling

[RC047]

Description

Adapt the authentication store with the related modules. This PR is a continuation from #127077

Motivation and Context

There are still some stores that use the old method, such as the ExposeAuthStore function, it might be better to adapt to current changes that no longer use this method.

Type of Change

• Dependency change (package changes such as removals, upgrades, or additions)
• Bug fix (non-breaking change that fixes an issue)
• New feature (non-breaking change that adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Non-code change (documentation, README, etc.)

Checklist

• My code follows the style guidelines of this project.
• All new and existing tests pass (npm test).
• Typings (e.g. index.d.ts) have been updated if necessary.
• Usage examples (e.g. example.js) / documentation have been updated if applicable.