No.
I have to check with my household insurance company whether the insurance liability also applies to my pet. This is independent of the status of the animal as ESA.
In many household insurance policies, certain animals are included in the liability package. In any case, an animal must also be reported when covering the insurance.
We therefore recommend that you consult with the insurance company and obtain detailed information about the coverage to property and personal injury. If necessary, you must take out your own pet insurance.
If your animal is then specifically admitted in a particular situation, all rules and regulations apply in addition to the on-site - because the consenting body also has certain partial liability obligations.