feat: Add Sentry error tracking to auth API functions (#5623)

## Summary
- Added comprehensive Sentry error tracking to all auth API functions
- Implemented helper functions to reduce code duplication  
- Properly distinguish between HTTP errors and network errors

## Changes
- Added `captureApiError` helper function for consistent error reporting
- Added `isHttpError` helper to prevent duplicate error capture
- Enhanced error tracking with:
  - Proper error type classification (`http_error` vs `network_error`)
  - HTTP status codes and response details
  - Operation names for better context
  - Route templates for better API endpoint tracking

## Test plan
- [ ] Verify auth functions work correctly in normal flow
- [ ] Test error scenarios (network failures, 4xx/5xx responses)
- [ ] Confirm Sentry receives proper error reports without duplicates
- [ ] Check that error messages are informative and actionable

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┆Issue is synchronized with this [Notion
page](https://www.notion.so/PR-5623-feat-Add-Sentry-error-tracking-to-auth-API-functions-2716d73d3650819fbb15e73d19642235)
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Author: viva-jinyi