I can speak from experience; my therapist decided that I would benefit from a pet to help manage my anxiety and depression. My cat that I have now helps me with my routine, provides comfort and therapeutic benefit to me, and is (generally) a calm and stable presence in my life (except when she gets the 4AM crazies).

My therapist said nothing of this in my accommodation letter; she simply stated that due to the nature of my disability, I was requesting accommodation by means of adopting a cat for emotional support. I had a long history with this doctor which was provided in my letter, to show that we have an established relationship and that this was not just a cheap trick to get myself a cat. I also make sure that I don't present her as a service animal in any way, because she's not. If a request is legitimate, and you don't act sketchy, there's no need to go above and beyond to explain the "why".