feat: implement M3 multi-browser support (Firefox, Edge, Safari)

Add browser abstraction layer (compat.ts) that normalizes Chrome/Firefox/
Edge/Safari extension APIs behind a unified promise-based interface.
Replace all direct chrome.* calls with compat layer across service worker,
sync engine, badge, storage, popup, and options modules.

- Browser-specific manifests in manifests/ (Firefox MV3 with gecko settings,
  Edge/Safari variants)
- Multi-target build system: `npm run build` produces all four browser
  builds in build/<browser>/
- Per-browser build scripts: build:chrome, build:firefox, build:edge, build:safari
- Auto-detects browser at runtime for platform-specific device registration
- All 38 existing tests pass, typecheck clean

Co-Authored-By: Paperclip <noreply@paperclip.ing>

extension/.gitignore

@@ -1,2 +1,3 @@
 node_modules/
 dist/
+build/