Do multiple taming attempts stack in BOTW?
Nope, the only mechanic to horse taming is that based on the specific horse stats / type you have to press the "soothe" button a specific number of times before running out of stamina. There is nothing else to that, and good button mashing skills will allow you to get every horse with as little as one stamina wheel.
You can get more info in this Youtube video by Croton. Generally speaking, the average horse requires 15 button presses. Royal stallion is 25, and the big black "Ganondor-style" horse requires 30.