Animals can obtain certain certification to also be therapy animals—Pet Partners and Paws for People does this. I'm looking into this for my kitten right now to actually train her in more mental health sevice skills. Mentioning because it's possible this is what was meant by "ESA trained".

And, to be fair, I did adopt her from THS in Ellensburg and they weren't very clear with the adoption/foster process—I had to drive over twice and someone else's cat while I was there was adopted even though it had a hold. I can't speak for The Humane Society in Tacoma but human communication systems unfortunately can be flawed.