Only if they are as registered as Nintendogs. There has been thousands of rip-offs of Mario and Pokémon over the years where they are basically the same with just a few exceptions and changing names and nothing happens.Oh? Are they? Answer me this then. Mario is a trademarked character. Link is a trademarked character. Pokemon are trademarked characters. You may use none of those characters in a video game of your own without getting permission first, lest you want to get into legal troubles. If you started selling your own unlicensed copies of "Mario's Brownie Bash" on Steam, oh boy, you'd be in some deep **** with Nintendo's legal team.
So tell me, since Dalmatians are a type of Nintendog, are you saying those are copyrighted too? Can I not use a dalmatian in my own game without crediting Nintendo first? After all, they are video game characters as much as Mario and Link are since Nintendogs has them. Like it or not, that is literally what you are implying if you say they're as much of a video game character as Mario or Link is.
You can't copyright an existing species of animal, meaning you can't copyright the breeds in Nintendogs. The brand is all Nintendo has control over. Any other 3rd party could just as easily make their own dog simulator on the PS Vita and have no issues getting it published and sold.
http://nintendogspluscats.nintendo.com/
My point is that, as trademarked as Nintendogs, they are videogame characters just like any other. If they have a generic design which is basically the same as real life counterpart is just another issue, but that doesn't make them any less Nintendo than others.