Is there any evidence about this being widely abused? Aside from anecdotal evidence? I would assume that even in the US airlines will require a doctors certificate and not just let any cat & dog board.

As for all the people shouting abuse: A member of my family had an ESA. While her symptoms allowed her to do many daily tasks without the animal she still made a choice not to travel by air (and in fact made many other compromises in daily life).

I can easily imagine situations where travel is non-optional and an ESA can make all the difference. This thinking of "guide dogs are fine but support animals for other disabilities are not" is missing the point of many mental illnesses.