The Singing Dog Series (シンギングドッグ シリーズ) is a series of singing, animated toy dogs. The dog with the black base appears to have been initially created by Yuen Shing Toys (YS Toys), but a company called Beverly planned on distributing the rubber dogs, pictured above, in Japan. Beverly's suggested retail price was 3,500 yen.
Another distributor called Eddy Toys sold it in Europe.
The name on this page was given by Beverly; only one official name, which was given by Eddy Toys to the dog with a black base, appears to have been documented. I don't know for sure if Yuen Shing was responsible the other two types of rubber dogs pictured, and I don't know if they got released.
The description for the photo above states the toys were in the development stage and that the final version may be different from what's pictured.
Though the first photo on this page shows a dog on a base that is mostly black with some brown (a dachshund?), this exact version is not confirmed to have been released.
Three released versions of the dog on the base exist, including a dalmatian, a dog that's mostly black with white as its secondary color, and some brown fur near its eyes, and a dog that's mostly brown with a black spot on its back and ears that are partially black. Yuen Shing Toys sold these dogs under the name "Singing Puppy".
The dog barks to two tunes that it plays. It can wag its tail, raise its body, and move its mouth while barking.
Credit to _ksllmfcai_ for mentioning the first two with videos.
Two bootleg plush versions, one featuring a yellow Moomin in a plush basket, and another which features a THE DOG plush in a plush basket, exist. The company that made these is also unknown.
There are no videos showing what this one does or what it sounds like, but it does raise itself and yap, according to the box. The box also has musical notes, which could imply this plays music.