As much as I would like to think so, even this isn't always the case. Owners can train their own service dogs and there are practically no standards other than "performs a task".
...As I recall a woman on an EZE/USA flight telling another passenger essentially how she games the system (while fawning over the "cute dog") so that she can travel everywhere with her dog.
While the airline is in a "no win" situation, the airline is also a proper defendant. Speaking generally, if you must get to Gate 100 to get to your flight on ABC Airlines, and "service animals" are permitted on your route to Gate 100, and a "service animal" attacks on your route to Gate 100, and you really had no other access to Gate 100, and it is known that the moniker "service animal" includes untrained bogus "service" animals, there's no question that the airline put you in foreseeable harm.
When you get to the "can I pet your dog" situation, the pool of liable defendants isn't so clear. On the other hand, airlines seem to encourage the "oh so cute" and so on, with onboard animals, even not having issue with them on passenger seats, petting them (those that should be in the carrier the entire flight), and so on ... this has got to stop. I write this with a big pouty dog lying on the floor right next to me. He's never been on a plane. He doesn't belong on one.