The cops in the video seem to be on the up-and-up. Also, if the owner took it back, you can bet BATF would be waiting to bust her. The PD will probably hand it off directly to a museum for her. In this case, they actually seem to be acting in this woman’s best interest.
I was wondering since its a machine gun in no working condition if she would now need a tax stamp to own it.
That's part of the training: "How to pretend you aren't an A-hole in front of cameras." They are very good at adopting a "WTF" expression when you try to recover something. "Gun? What gun? Nobody here signed that reciept."
But I do believe you're correct that if she tried to take it home, the BATFE would have busted her for sure.
I know an ATF agent. He was actually sickened the day he was required by law to confiscate a similar item form a citizen. His choice was to destroy it or ensconce in in the ATF private museum. Selling to a dealer or giving it to the National Firearms museum was not an option.