♻️(backend) sync lobby and LiveKit participant UUID generation

Refactor lobby system to use consistent UUID v4 across lobby
registration and LiveKit token participant identity instead of
generating separate UUIDs.

Maintains synchronized identifiers between lobby cache and LiveKit
participants, simplifying future participant removal operations by
using the same UUID reference across both systems.

src/backend/core/utils.py

@@ -56,6 +56,7 @@ def generate_token(
     color: Optional[str] = None,
     sources: Optional[List[str]] = None,
     is_admin_or_owner: bool = False,
+    participant_id: Optional[str] = None,
 ) -> str:
     """Generate a LiveKit access token for a user in a specific room.
 
@@ -69,6 +70,8 @@ def generate_token(
         sources: (Optional[List[str]]): List of media sources the user can publish
                          If none, defaults to LIVEKIT_DEFAULT_SOURCES.
         is_admin_or_owner (bool): Whether user has admin privileges
+        participant_id (Optional[str]): Stable identifier for anonymous users;
+                         used as identity when user.is_anonymous.
 
     Returns:
         str: The LiveKit JWT access token.
@@ -90,7 +93,7 @@ def generate_token(
     )
 
     if user.is_anonymous:
-        identity = str(uuid4())
+        identity = participant_id or str(uuid4())
         default_username = "Anonymous"
     else:
         identity = str(user.sub)
@@ -120,6 +123,7 @@ def generate_livekit_config(
     is_admin_or_owner: bool,
     color: Optional[str] = None,
     configuration: Optional[dict] = None,
+    participant_id: Optional[str] = None,
 ) -> dict:
     """Generate LiveKit configuration for room access.
 
@@ -130,6 +134,8 @@ def generate_livekit_config(
         is_admin_or_owner (bool): Whether the user has admin/owner privileges for this room.
         color (Optional[str]): Optional color to associate with the participant.
         configuration (Optional[dict]): Room configuration dict that can override default settings.
+        participant_id (Optional[str]): Stable identifier for anonymous users;
+                         used as identity when user.is_anonymous.
 
     Returns:
         dict: LiveKit configuration with URL, room and access token
@@ -149,6 +155,7 @@ def generate_livekit_config(
             color=color,
             sources=sources,
             is_admin_or_owner=is_admin_or_owner,
+            participant_id=participant_id,
         ),
     }