I read a while back (did not independently confirm) that they used a DJI Inspire for most of the promo video footage. I think the Inspire can do a lot of the same stuff too.
I thought it was funny they were using an existing product from a competitor to simulate what they claimed their future product could do.
It seems the Mavick tracks objects by image recognition from the camera. I'm not sure the stuff shown on the Lily video where you track a presumably radio transmitter from a small device is actually a thing that exists. I was trying to figure out how to do it for tracking kids if they run off in stores etc and it's not easy. You could probably do it with a several feet across receiver so you could detect the timing differences.
In the Lily faked video the Inspire was flown by a human operator.
Guess it the tracker and drone both had gps receivers and communicated that could work although it still might be hard to get the altitude right.
That said, it looks like they're not shipping the second version anymore and are taking preorder money for the 3rd version so this company could be under duress as well. But the tech exists and is at least somwhat commercially viable.
Also it seems to just rotate a camera tripod mount and so doesn't have the altitude problem. Having mucked about with gps the lat and long are fairly accurate - typically off by 10m whereas gps altitude is often off by 200m which is easily enough to send your drone into the ground. I guess if it did differential timing between the two gps's that could work or they could measure ground altitude some other way. Dunno if Lily cracked that. Conventional aircraft in fog use barometric altimeters corrected for weather conditions. And that's only accurate to ~100ft.
I didn't quite crack the finding kids in a shop problem. Satellite gps wouldn't do it but I was thinking maybe ultrasonics - radio signal makes the tracker beep then you could use loudness / timing to a few microphones a few inches apart like human hearing locating sounds. Not sure if anyone's built something like that.