(WIP) Add SDK specificity across the docs for the Customizing Clerk section

[SarahSoutoul]

🔎 Previews:

• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/appearance-prop/overview
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/appearance-prop/layout
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/appearance-prop/themes
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/appearance-prop/variables - missing Using CSS variables section
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/appearance-prop/captcha
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/adding-items/organization-profile (missing JS for last example)
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/adding-items/user-profile (missing JS)
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/adding-items/user-button (missing JS)
• https://clerk.com/docs/pr/ss-sdk-specificity-project-customizing-clerk/guides/customizing-clerk/localization

What does this solve?

This PR is part of the overall SDK-specificity project, but specifically targets the pages within the Customizing Clerk section.

What changed?

• Added code examples for all necessary SDKs
• Removed Remix code examples

Checklist

• I have clicked on "Files changed" and performed a thorough self-review
• All existing checks pass

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Project	Deployment	Preview	Updated (UTC)
clerk-docs	Ready	Preview	Nov 3, 2025 11:43pm