As of January 7th, 2019, United Airlines have banned ALL emotional support animals on long haul flights. United Airlines has recently decided to forbid animals on flights under the influence of Delta. Delta is their biggest competitor who prohibited puppies and kittens on board in December 2018.
United is only allowing emotional support cats and dogs to attend flights no longer than eight hours. Puppies and kittens under four months are not permitted to travel on any flights. The new enforcement is effective from January 7th, 2019. However, United has made allowances for passengers travelling with animals who booked before January 3rd.
However miniature horses and cats and dogs acting as service animals will be accepted on board.
Support and service animals can fly for free in the States under the 1986 Air Carrier Access Act.
Due to the growth of animals now attending flights with passengers, airlines have incurred several incidents including soiled cabins, attacks and allergies. It is said that an increase of these events is forcing airlines to impose these rules.
United Airlines Stated: “We have seen increases in onboard incidents on longer flights involving these animals, many of which are unaccustomed to spending an extended amount of time in the cabin of an aircraft.” (https://www.independent.co.uk/travel/news-and-advice/united-airlines-emotional-support-animals-banned-flights-delta-puppies-kittens-a8710901.html)
Delta stated daily they flew 700 animals with passengers including snakes and spiders.
A United Airlines passenger attempted to board to Los Angeles with an emotional support peacock named Dexter in 2018. She even tried to purchase a ticket for him, however he did not satisfy the weight and size guidelines, so she was refused.
International airlines outside the U.S.A. do not recognise emotional support animals. However, service animals assisting disabled passengers are permitted to travel usually without any additional cost.
As regards Wayne Rooney flying into Dulles, well that was an entirely different emotional support package.