So I have a dog and am therefore not allergic to anything. Maybe some humans.
IMHO :
True service dogs are just that - identified by a vest, well behaved, never off leash in a public place including a lounge or on the plane. They do not stray from their owner more than a very short leash distance.
Then there are ESAs. No doubt some people need them. These are house pets that provide a comfort factor. These tend not be be trained to the level of service dogs. They will bark, lick everything, play and wander off. Hence these need to be on leash or crated at all times in an airport or on a plane. Having them wandering down the plane aisle or in a lounge is a no no.
Then there are the fakes in both categories with ESA fakes more rampant specially in the US where I fly at least 2 times a month.
I doubt the airlines can do much about identifying fakes since they would have to accept the relevant paperwork. This should fall on aviation authorities to set more rigid rules. What the airlines should do is clamp down on behavior. If people let dogs wander around a plane or a lounge, ban them from traveling with a animal as they obviously cannot control the animal or are disregarding the rules. Provide a note to every passenger with an animal with a behavior expectation - on leash, crated, etc. We all follow rules at the airport and on the plane and so should they.