Even if it was a dog or other normal pet, an actual registered ESA doesn’t allow you to take them to public things. It just allows you to keep them in your domicile when it normally wouldn’t. That’s pretty much the only actual benefit of it. It doesn’t require the animal to be trained, and doesn’t really mean shit.
Source: My husky is a registered ESA.
What can be allowed in public spaces are PSAs. These are harder to register as the owner of the animal actually needs to have a very good reason to have one, the animal must also be trained, and you can only get started in the first place (for either a PSA or ESA) with the recommendation of a psychiatric professional.