well there are lots of different consoles and systems with horse games with no breeding, to answer fully we would need to know what system your looking at.
Wolf Play
yes virtual bark is
A Virtual Horse AVH
foopets.com
There's AVH and Howrse ( AVH stands for A Virtual Horse)
Free games in every category can now be found online. For free horse riding games, try Horse Games, Horse Riding Games, or Mary. Most of these sites only require a free registration to play.
wolf quest its really cool i hope this helped also dog and cat breeding games are foopets
there is bellasara.com,avirtualhorse.com and thats all i know of
foopets.com
On Horseland you can breed horses and dogs. For Nintendo ds the is a game called nintendogs
Try Wajas.com.
I also know myhorseclub, but you are riding the horse, then animal jam, howrse, starstable, horseland. Hope it help ;))