♻️(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/services/lobby.py

@@ -89,7 +89,7 @@ class LobbyService:
     @staticmethod
     def _get_or_create_participant_id(request) -> str:
         """Extract unique participant identifier from the request."""
-        return request.COOKIES.get(settings.LOBBY_COOKIE_NAME, uuid.uuid4().hex)
+        return request.COOKIES.get(settings.LOBBY_COOKIE_NAME, str(uuid.uuid4()))
 
     @staticmethod
     def prepare_response(response, participant_id):
@@ -163,6 +163,7 @@ class LobbyService:
                 color=participant.color,
                 configuration=room.configuration,
                 is_admin_or_owner=False,
+                participant_id=participant_id,
             )
             return participant, livekit_config
 
@@ -183,6 +184,7 @@ class LobbyService:
                 color=participant.color,
                 configuration=room.configuration,
                 is_admin_or_owner=False,
+                participant_id=participant_id,
             )
 
         return participant, livekit_config