Add storage-based state synchronization for extension components

Replace direct port communication with chrome.storage API for state sync
between devtools, background, content, and injected scripts. Add storage
manager, state sync utilities, and webextension-polyfill dependency.
Update manifest to use module service worker.

Author: dborysov

memory-bank

@@ -32,7 +32,8 @@
     "lucide-react": "^0.540.0",
     "react": "^19.1.0",
     "react-dom": "^19.1.1",
-    "superjson": "^2.2.2"
+    "superjson": "^2.2.2",
+    "webextension-polyfill": "^0.12.0"
   },
   "devDependencies": {
     "@eslint/js": "^9.33.0",

package-lock.json

@@ -15,7 +15,8 @@
     }
   ],
   "background": {
-    "service_worker": "background.js"
+    "service_worker": "background.js",
+    "type": "module"
   },
   "web_accessible_resources": [
     {