♻️(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/tests/services/test_lobby.py

@@ -144,10 +144,9 @@ def test_get_or_create_participant_id_from_cookie(lobby_service):
     assert participant_id == "existing-id"
 
 
-@mock.patch("uuid.uuid4")
+@mock.patch.object(uuid, "uuid4", return_value="generated-id")
 def test_get_or_create_participant_id_new(mock_uuid4, lobby_service):
     """Test creating new participant ID when cookie is missing."""
-    mock_uuid4.return_value = mock.Mock(hex="generated-id")
     request = mock.Mock()
     request.COOKIES = {}
 
@@ -268,6 +267,7 @@ def test_request_entry_public_room(
         color=participant.color,
         configuration=room.configuration,
         is_admin_or_owner=False,
+        participant_id="test-participant-id",
     )
 
     lobby_service._get_participant.assert_called_once_with(room.id, participant_id)
@@ -306,6 +306,7 @@ def test_request_entry_trusted_room(
         color=participant.color,
         configuration=room.configuration,
         is_admin_or_owner=False,
+        participant_id="test-participant-id",
     )
 
     lobby_service._get_participant.assert_called_once_with(room.id, participant_id)
@@ -400,6 +401,7 @@ def test_request_entry_accepted_participant(
         color="#123456",
         configuration=room.configuration,
         is_admin_or_owner=False,
+        participant_id="test-participant-id",
     )
     lobby_service._get_participant.assert_called_once_with(room.id, participant_id)