Example specialties are falconry and specific animals. Am I wrong or would a specialty birds apply wherever falconry does?
"Birds" isn't a specific animal but a wide category. "Horses" for a stablemaster or "sheep" for a farmer or "cows" for a Hibernian magus all count as they apply to one specific animal.
As "falconry" is valid, you may be able to argue with your storyguide that several closely related animals count, so "birds of prey" or "canines" or "mustelids" count, but anything wider than what is acceptable for a minor magical focus in Animal is likely to be too wide.
Note that falconry is different from "birds of prey". It probably does not tell you much about birds of prey in their "natural habitat". At the same time, it probably does provide information about birds and other small animals who are prey to your hunting bird.
I take the example "falconry" to mean "using/relating to animals in a specific type of activity". For example, I'd assume that "droving" (moving livestock over long distances, often with the help of dogs) would be a fair animal handling specialty. It covers moving sheep, cows, horses, camels etc. and handling of drover dogs. But it would not assist you in milking a cow or a sheep, nor in training dogs as attack dogs.