I admit ignorance here. The joke was funny even for the normal definition of Service Dog, adding the nuance of the ‘exchange for another dog’ and the dual/triple meaning of ‘service’.
People may have misunderstood the question as to if any regular service dog can come to services, but I don’t know of any other meaning than the official definition – A service dog is a type of assistance dog specifically trained to help people who have disabilities including visual difficulties, hearing impairments, mental illness, seizures, diabetes, autism, and more.
I think we are all better off if you DON’T explain what you think he meant.
BTW – I believe that a Guide dog is the same or maybe just a subset of the general term ‘Service dog’. “Service animals are defined as dogs that are individually trained to do work or perform tasks for people with disabilities. Examples of such work or tasks include guiding people who are blind”