No, that's just a travel cage - they have a much bigger cage. Their cages are normally in my room, they were moved into the studio only for the video and it was easier to just move the travel cage than their actual cage. The reason Zoey hangs out on their cage is because it hangs from the ceiling, thus it is "higher" than her cage.
I don't worry Zoey is going to fly away, for one she is trained to come on command. And two, I can barely keep her off of me as it is. She spends most of her time on my shoulder. While she loves to be outside on my shoulder, she actually doesn't like to fly outside.
The reason I free flighted her was so I wouldn't have to worry about her flying away. Flying outside is different than flying inside, there is wind which provides lift. If an inside bird does get outside, the wind takes it higher and higher and it can't fly down - thus, it continues to fly off until it is too tired to flap it's wings any longer and crash lands, usually about a mile away.
By training Zoey to fly outside in the wind, I don't have to worry about her flying off because she can't control her flight outside. She has no desire to run away, so there is really no need to worry about that. I worry more about hawks than anything else, and am always on the lookout for them.