Most animals do cause difficulty for their owners. You need to feed and take care of them. If they are not properly secured they tend to run away. You need to make arrangements for them when you can’t take them with you. While most of this is ignored by the GM, it still happens. Most fantasy setting expect people to use horses and the settings provided for them. Most if not all inns have a stable attached, or at least nearby. So in that sense the horses are DNPC’s.
A falcon is more exotic and unusual so there may not be the same level of support for it as there are for horses. With a horse you can simply turn it over to the stable boy and have him feed and take care of the horse, with a falcon this is probably not the case. The extra effort needed to take care of the falcon could be the complication. If it is a 0 pt complication it should not be that much of a burden.
It really depends on how the player wants to do it. Using the example of the war horse if the player wants a horse pretty much straight out of the book than it should be treated like equipment. If he wants something like Silver out of the Lone Ranger, that would be a follower. If he wants something slightly more powerful than the standard horse he could do it as a DNPC. It all depends on what the player wants.