I understand that certain aspects of owning a service dog are tax deductable- but what about therapy dogs?
There are no deductions or tax breaks of any kind for owning a therapy dog.
Some of the expenses of some types of service dog can be deducted as unreimbursed medical expense. Service animals are considered "durable medical assistive devices," the same as a wheel chair or walker and that's what makes their training and maintenance a medical expense. The IRS does not recognize all types of service dogs, including owner-trained dogs without documentation, and medical alert/response dogs.
A therapy dog used for visiting hospitals and nursing homes would not be medical equipment for you, so it would not qualify for such a deduction.
I doubt the therapy organization you are with gets a tax deduction. They probably get a tax exemption (to make purchases without paying sales tax), not from being a therapy dog group, but from being a 501(c)3 public charity.
There's a lot of confusion among the public between therapy dogs and service dogs. They are two entirely different things. Many dogs have jobs (therapy dogs, police dogs, search and rescue dogs, hunting dogs, etc.). Only service dogs are addressed under federal and state disability laws, such as the Americans with Disabilities Act. Service dogs may accompany their disabled handler to places where pet dogs are not permitted. Therapy dogs cannot. Service dogs are granted an exception to stay in rental housing and in hotels that do not permit pets and are exempted from deposits and pet fees, because they are medical devices, not pets.
Legally therapy dogs are considered pets. They have no more rights than pets do and their owners have no more rights than other pet owners. Therapy dog certification shows facilities such as hospitals and nursing homes that the dog has received certain training, temperament evaluations, health screening and has liability insurance. It is not a guarantee that the therapy dog can enter any hospital the owner chooses. They may enter only with permission from the hospital administration. The reasons for getting certified are to get that liability insurance and to get documentation to use in convincing the hospital to let you visit their facility.