Fix comparison badge to update in real-time across components

- Move comparison state from hook to shared context provider
- All components now share the same state instance
- Badge count updates immediately when schools are added/removed
- Add key prop to badge to re-trigger animation on count change
- Add storage event listener for cross-tab synchronization

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

nextjs-app/context/ComparisonContext.tsx

@@ -18,6 +18,7 @@ interface ComparisonContextType {
   clearAll: () => void;
   isSelected: (urn: number) => boolean;
   canAddMore: boolean;
+  isInitialized: boolean;
   mutate: () => void;
 }