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/components/Navigation.tsx

@@ -48,7 +48,9 @@ export function Navigation() {
           >
             Compare
             {selectedSchools.length > 0 && (
-              <span className={styles.badge}>{selectedSchools.length}</span>
+              <span key={selectedSchools.length} className={styles.badge}>
+                {selectedSchools.length}
+              </span>
             )}
           </Link>
           <Link