While the name given by the bootleggers is unknown, and the toy company responsible for these is also unknown, what we do know is they can open and shut their mouths while playing a pre-recorded audio clip. The eBay seller for the one pictured above states this one plays "Rock-A-Bye-Baby".
I don't know if all of them have the trembling mechanism, as the eBay listing for the first one on this page does not mention it trembles, but there's a video on YouTube showing one vibrating while moving across the floor, and its cheeks having flashing lights. While the motor is very loud in the video, it can be heard saying "Happy Birthday". Only thing is, the mouth isn't moving though it should.
The one in the first image on this page does not appear to have glowing cheeks.
The last pictures in the gallery are of the one in the video, and came from the listing.