feat(auth): support multi-profile Plex Home login with PIN

[0xSysR3ll]

Description

#1594 ended up being a bit too large to review comfortably, so it has been split into three smaller, more focused PRs.
This is the second PR in the stack. It builds on #3271 and adds the multi-profile Plex Home login flow with PIN support.

• Fixes Add plex home linking #1402
• Closes feat(auth): support Plex home profile login #1594

How Has This Been Tested?

Preview image, see #1627

Screenshots / Logs (if applicable)

Checklist:

• I have read and followed the contribution guidelines.
• Disclosed any use of AI (see our policy)
• I have updated the documentation accordingly.
• All new and existing tests passed.
• Successful build pnpm build
• Translation keys pnpm i18n:extract
• Database migration (if required)

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[github-actions]

This pull request has merge conflicts. Please resolve the conflicts so the PR can be successfully reviewed and merged.

[github-actions]

This pull request has merge conflicts. Please resolve the conflicts so the PR can be successfully reviewed and merged.