Look like Language Studio only works with "language Service". Can't find it explicitly written for multi-service but based on this page, look like it's a service to be used with "Azure Cognitive Service for Language":
https://learn.microsoft.com/en-us/azure/cognitive-services/language-service/language-studio
This page suggest migrating to "Azure Cognitive Service for Language"
https://learn.microsoft.com/en-us/azure/cognitive-services/language-service/concepts/migrate