It is only called hoarding when it is done compulsively. Many people have loads of animals and/or birds etc. and it become so much that they cant look after them properly. They buy on impulse without any planning or research.
They even buy the animal / bird before buying and setting up the cage etc.
If you carefully plan, research and set up the cage etc. and then go and find the one that you feel attracted to, then I would not call it hoarding.
I actually saw that program. It was quite disturbing. The house was full of birds, other stuff??, papers, to the point where it was impossible to move around. No normal person could live in the house, husband had moved out.....Very sad!