chore(docs): update `migrating-to-v5` docs

[halvaradop]

☕️ Reasoning

This pull request updates the documentation in the migrating-to-v5 section by expanding and improving the existing content. A new Route Handlers tab was added to the details section, providing guidance on how to use NextAuth.js v5 with Route Handlers. Additionally, the Client Component section was revised to address multiple issues raised by the community regarding unclear documentation.

The format:write script was also run to ensure consistent formatting across the documentation files.

It’s been a while since my last contribution to the project, so I’m glad to be back and help improve the docs.

Key Changes

• Updated the Client Component section
• Added a new Route Handlers section
• Ran the format:write script

🧢 Checklist

• [x] Documentation
• [ ] Tests
• [ ] Ready to be merged

🎫 Affected issues

#12767

📌 Resources

• Security guidelines
• Contributing guidelines
• Code of conduct
• Contributing to Open Source

[vercel[bot]]

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Project	Deployment	Preview	Comments	Updated (UTC)
auth-docs	Ready	Preview	Comment	Aug 24, 2025 10:46pm

1 Skipped Deployment

Project	Deployment	Preview	Comments	Updated (UTC)
next-auth-docs	Ignored	Preview		Aug 24, 2025 10:46pm

[vercel[bot]]

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[codecov[bot]]

Codecov Report

:white_check_mark: All modified and coverable lines are covered by tests. :white_check_mark: Project coverage is 37.89%. Comparing base (39dd3b9) to head (202b3ee).

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #12872      +/-   ##
==========================================
- Coverage   38.76%   37.89%   -0.87%     
==========================================
  Files         200      195       -5     
  Lines       32323    31137    -1186     
  Branches     1393     1298      -95     
==========================================
- Hits        12529    11799     -730     
+ Misses      19794    19338     -456     

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