Too late for me to edit my earlier post, but I wanted to further clarify this.
No one is making the claim that you own your falconry bird because it resides on (or in) your private land [home]. You own it because you rendered it into private possession at the time you lawfully bought or trapped it. And in much the same way, you relinquish your ownership when you release it back to the wild - although the way the federal falconry regulations are spelled out, they clearly recognize your ownership even after that event because you can retrap your bird at any time with absolutely no time limits.