This was inspired by the question How would tattoos fare on reptilian scales Let's assume for this question that humans never lost their hairy, hairy bodies and are covered in fur. How would tattooing work? Would they use scarification, or maybe freeze branding?
With thin hair, such as horses have, obviously, branding could be a preferred option. It would depend on your creature's design.
As an alternative to tattoos, distinctive patterns can be continually trimmed as identifying symbols in your animal. This works for simple patterns.
This can be a temporary alternative to your tattoos, by dying the furs. However, I think the following would be most what you're looking for.
Shaving + Tattoo
Shave the area, apply the tattoo, and then continually keep that part of the area shaved, so the drawing and pattern will show when you want it to.
One possibility is quite... nasty, and as such it would probably not be used for "stetic only" tattoos but just for cultures that use those tattoos as a way of identification / tribal identity.
The word here is scarification; produce designs out of wounds on the skin. In this case, since you want the affected area to be easily visible, you would look for a part of the body which already has little hair on it (shoulders) and mark it with identifiable shapes. As you would want a relatively big patch of skin to be affected (so it does not get covered easily), you probably will want burning it instead of other methods (like the one with the Euler formula in the article).
An example could be similar to the rank insignias used by the military, with more "bars" of cleaned skin marking a braver warrior or a more important chief.
Once that is done, it is up to experimentation to see if the affected area can be colored with more traditional tattoos.
With time (and a more advanced technology), more cosmetic "tattos" could be accomplished by using depilation.
Since an injury can make the hair grow back a different color due to the skin trauma, one might use that process to create permanent hair color patterns. Perhaps they would experiment with different sorts of injuries which might create different shades of color.
Also, how about hair grafts to create color patterns. They might even have adapted scalping and fused it with grafting. Multicolored individuals would be great warriors...