The Papillon and Maltese breeds were in development for the game but never finished. Their placeholder files can still be found in the game's code. The Maltese did eventually become a selectable breed in Nintendogs + Cats.
Contributed by KidDivinegon
The Boxer, Siberian Husky, Golden Retriever, and Dalmation breeds are not in the original Japanese versions of the game. They were added in the localized versions. These breeds would later be included in Japanese regions for the first time in Nintendogs + Cats.
Contributed by VGTurtle
A Wii version was considered, but it was ultimately decided that the voice recognition, utilizing the DS's integrated microphone, was too integral to the experience. Because the Wii did not have a microphone, the player's dog would not be able to respond to their voice.

It was apparently decided that it would not merit the creation of an external peripheral, such as the Nintendo 64's Voice Recognition Unit required for Hey You, Pikachu!, despite the creation of the Wii Speak accessory alongside Animal Crossing: City Folk in 2008.
Contributed by Spherix
During development, Nintendo had considered other animals for the game that became Nintendogs, as the preferences of the development team varied, with the main programmer being "more of a cat person". Shigeru Miyamoto's decision was influenced by his own experiences with the family dog, a Shetland Sheepdog named Pikku:

"The reason it ended up being a dog game is because about four years ago me and my family actually got our first dog"
Contributed by Spherix
There are actually 5 different versions of Nintendogs released. In chronological order, Daschaund and Friends, Lab and Friends (Known as Shiba and Friends in Japan), Chihuahua and Friends, Best Friends, and Dalmatian and Friends.
Contributed by Shadowmane