feat: Remove localStorage for game states and implement cross-domain player ID sync

- Remove localStorage for game states and statistics (backend only)
- Add API route to suggest player ID based on recently updated states
- Add async player ID lookup that finds existing IDs across domains
- When visiting a new domain, automatically find and use existing player ID
- Enables cross-domain synchronization between hoerdle.de and hördle.de

lib/playerId.ts

@@ -20,11 +20,75 @@ function generatePlayerId(): string {
     });
 }
 
+/**
+ * Try to find an existing player ID from the backend
+ * 
+ * @param genreKey - Genre key to search for
+ * @returns Player ID if found, null otherwise
+ */
+async function findExistingPlayerId(genreKey: string): Promise<string | null> {
+    try {
+        const response = await fetch('/api/player-id/suggest', {
+            method: 'POST',
+            headers: {
+                'Content-Type': 'application/json',
+            },
+            body: JSON.stringify({ genreKey }),
+        });
+
+        if (response.ok) {
+            const data = await response.json();
+            if (data.playerId) {
+                return data.playerId;
+            }
+        }
+    } catch (error) {
+        console.warn('[playerId] Failed to find existing player ID:', error);
+    }
+    return null;
+}
+
 /**
  * Get or create a player identifier
  * 
- * If no identifier exists in localStorage, a new UUID is generated and stored.
- * The same identifier is used across all domains (hoerdle.de and hördle.de).
+ * If no identifier exists in localStorage, tries to find an existing one from the backend
+ * (based on recently updated states). If none found, generates a new UUID.
+ * This enables cross-domain synchronization between hoerdle.de and hördle.de.
+ * 
+ * @param genreKey - Optional genre key to search for existing player ID
+ * @returns Player identifier (UUID v4)
+ */
+export async function getOrCreatePlayerIdAsync(genreKey?: string): Promise<string> {
+    if (typeof window === 'undefined') {
+        // Server-side: return empty string (not used on server)
+        return '';
+    }
+    
+    let playerId = localStorage.getItem(STORAGE_KEY);
+    
+    if (!playerId) {
+        // Try to find an existing player ID from backend if genreKey is provided
+        if (genreKey) {
+            const existingId = await findExistingPlayerId(genreKey);
+            if (existingId) {
+                playerId = existingId;
+                localStorage.setItem(STORAGE_KEY, playerId);
+                return playerId;
+            }
+        }
+        
+        // Generate new UUID if no existing ID found
+        playerId = generatePlayerId();
+        localStorage.setItem(STORAGE_KEY, playerId);
+    }
+    
+    return playerId;
+}
+
+/**
+ * Get or create a player identifier (synchronous version)
+ * 
+ * This is the legacy synchronous version. For cross-domain sync, use getOrCreatePlayerIdAsync instead.
  * 
  * @returns Player identifier (UUID v4)
  */