I am in agreement with quite a few of the trainers. That is an unusual request. I actually can add trainers to my insurance but I would never ask a trainer to add my facility to their insurance. I agree with Harris, I might be worried that the facility might not have insurance.
I’m with Karin on this one. Assuming you are an employee, not an independent contractor,then your employer (i.e.the studio owner) should provide insurance. As a safeguard, you probably should have professional liability insurance but that is an issue to discuss with your insurance agent. If you are working as an independent contractor,the studio owner is not your employer, but is contracting your services, therefore the owner may require that the business (i.e.the studio) be listed under your insurance for liability purposes. This is a reasonable and legitimate request to protect the interests of the studio owner.
you should talk to your insurance company about this. I find this an unusual request because usually it is the studio that has the insurance, and the instructors are covered under their policy. This is even more so as you say that you have been hired by the studio, thus being an employee. I would ask them whether they have insurance.