In the United States if you have severe animal allergies, first it's important to chose a flight that doesn't allow pets on board. In the US, allergies of others is not a valid reason to deny entry to service animals however, because US law recognizes that animal allergies rarely reach a level of constituting a disability.
However, if you are a rare person who has actually life threatening allergies to dogs where you end up on the hospital- when you book your flight you need to mark that you have a disability and you need special assistance. You should write in the comments you have life threatening animal allergies and in the case of any service animals on board need to be seated as far from the animal as possible. Then follow the airlines instructions for disabled passengers - you may also need to contact a special hotline with the airline to make sure of this and make sure to bring the documentation that you will need to be reseated on board.
As someone myself who has life threatening airborne allergies to some foods, understand how difficult that makes flying. You should also seriously consider wearing a well fitting N-95, and pre dosing for flights taking your antihistamines and allergy meds and steroids if needed before high risk situations and ensure you have adequate inhalers epipens and antihistamines on board. People with mild allergies, who have itchiness etc can also consider wearing a well fitting n95, and taking allergy medicine, and seeing at the time of the flight if someone would be so kind as to switch seats with you. But for a mild allergy it may not have to be accommodated. I have had to use epipens on planes so I get it- you should also think about seeing your doctors before you travel given your life threatening allergies/asthma to make a plan for emergencies
Unfortunately sometimes two people with disabilities need exactly opposite things. For example: strobe fire alarm lights can save lives of Deaf folks in a fire, but can also cause seizures in photo sensitive epilepsy.
