I've got food allergies, and I'm married to someone with several animal allergies. The fact is, when you are the one with the known allergy, it is YOUR responsibility to be proactive in avoiding the allergen, and your responsibility to carry appropriate medication in any situation where you will not be able to access immediate medical assistance. If it is a serious issue for you, you ask -- every time. In the case of an air flight, you ask when you buy the seat.

As to who changes flights, I say the first person to warn the airline gets to stay. If you tell the airline you are bringing an animal, then the airline should mark the flight as containing one. If you are allergic, you ask and avoid that flight. If you are allergic and tell the airline so, and no one has so far declared a pet, then the flight should be marked to not allow pets on that run. (This is already done with nuts.)

In the OP, the FA should have made a general announcement once the passenger made the situation known. She did not have to announce what the allergen was, but simply say something like, "we have someone in row 28 who is having a rather severe allergic reaction, and needs to change seats. If you are seated forward of row 22 and might be willing to switch with her, please ring your call bell." Then the FA could explain what the reaction is, and the volunteer could at that time decide whether or not to do it.