feat(clerk-js,ui,react): Port enable orgs prompt to core 3

[LauraBeatris]

Description

Backport changes for the "enable orgs" prompt - from #7159, #7307, #7298, #7334

Checklist

• pnpm test runs as expected.
• pnpm build runs as expected.
• (If applicable) JSDoc comments have been added or updated for any package exports
• (If applicable) Documentation has been updated

Type of change

• 🐛 Bug fix
• 🌟 New feature
• 🔨 Breaking change
• 📖 Refactoring / dependency upgrade / documentation
• other:

[changeset-bot]

⚠️ No Changeset found

Latest commit: 55d0ab0

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✨ Finishing touches

🧪 Generate unit tests (beta)

• Create PR with unit tests
• Post copyable unit tests in a comment
• Commit unit tests in branch laura/backport-enable-orgs-prompt

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Project	Deployment	Preview	Comments	Updated (UTC)
clerk-js-sandbox	Ready	Preview	Comment	Dec 2, 2025 4:08pm

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commit: 55d0ab0