It's all the same dog, and it's basically just a goof.
The dog following them to Lebowski's house after Bunny comes back is Walter's dog according to the official script (and also in this script online).
The Dude and Walter nevertheless continue toward the great study. Walter's dog follows, stiffly wagging its tail.
And we know it's Walter's dog because it's in the van when Walter picks the Dude up. We can hear the dog barking in the van and briefly see it sticking its head out the window in the background when the Dude is talking to the private detective. It also follows Walter around at Lebowski's house, and it obviously looks the same.
So that previous scene is basically just a goof, and IMDb and Moviemistakes.com seem to concur. But I'm saying 'basically' because it could be explained, at a stretch: When we see Walter placing the empty cage on the roof of the car, opening the door and putting his bowling ball inside, we hear what sounds like dog claws shuffling and its tag rattling before Walter closes the door. Though he never looked down, he held the door open long enough for the dog to jump in (we just didn't see it), and probably didn't bother to put it in its cage just for the ride home.