1) Why are you assuming this is an ESA?
2) the airline DOES have recourse here, the law allows them to require people to buy extra seats if the passenger and their dog doesn't fit into the space allocated for a single seat.
1) The dog licked the OP. True service animals are trained not to behave this way.
2) It shouldn't be left up to the seatmate to complain and make a fuss. Airline staff should proactively solve the problem, including by insisting that the animal's owner buy a second seat on a later flight if necessary. It shouldn't have been the OP's burden.