feat: improve cross-browser extension compatibility (#204)

[kyoungholee]

Summary

• Cleaned up Chrome-specific fields in the extension manifest to align with the WebExtension spec.
• Removed the unused identity permission so the extension is easier to load in other browsers.
• Prepared the manifest so the same build can be used for Safari/Firefox packaging.

Changes

• Updated client/generate-manifest.js:
  • Removed the Chrome-only key field from the generated manifest.
  • Changed extension permissions to only use storage.
• Updated client/public/manifest.json:
  • Removed the hardcoded key field.
  • Updated permissions from ["identity", "storage"] to ["storage"].

Rationale

• Avoids Chrome-only fields that can cause validation or review issues in Firefox/Safari.
• Follows WebExtension best practices for cross-browser compatibility.
• Makes it easier to distribute the same extension code across multiple browsers.

Testing

• npm run chrome:generate-manifest in client and confirmed public/manifest.json is generated without key and with ["storage"] permission only.
• npm run chrome:build in client to ensure the extension build still succeeds.