As per the documentation (https://help.salesforce.com/articleView?id=omnichannel_supervisor_change_queues_skills.htm&type=5), Omnichannel Supervisor can change the Agent's queue using the "Change Queues" button. But I don't see that option in the Service console (Classic). Is this feature available only for Lightning?
That's correct, the feature is only available in Lightning.
The use the feature, a Supervisor also needs to have the following permissions:
- Customize Application
- Manage Public List Views and Manage Users
To change skills:
- Create, Read, Update on service resources
- Read Access and Edit Access for the Active field on Service Resource Field-Level Security
- Read Access and Edit Access for the following fields on Service Resource Skill Field-Level Security: End Date, Start Date, and Skill Level