Hamish,
Your tarantula is not a service animal unless it can demonstrate that is can perform a specific service that is needed for daily life and a licensed physician has written a letter detailing that it is trained and needed to help due to a physical disability.
Your tarantula can be a emotional support animal (ESA) if you have a letter from a licensed physician or psychiatrist that it provides you with emotional support. But, that does not give you the legal right to take it everywhere, only places where ESA's are allowed by law (certain housing) or if it is an airline and you meet their requirements for a ESA.
AAT animals are animals who have training to provide "comfort", usually dogs and cats, and they have a owner or handler who brings them to hospitals, senior centers, or where ever the person or people are who need comfort. The dogs or cats are trained to be petted, held and do not remain there. Most recently, AAT dogs were used to comfort kids in shelters after the hurricane