I have problems with acid reflux, and have suffered severe tooth erosion and some esophagus damage over the years. In my case, bringing up stomach contents is accidental. Whoever, a ruminant brings up a load of cud several times a day. Wouldn’t that cause similar issues? Why not?
The cud the ruminants bring up is the stuff in the rumen, which is kept, in normal conditions, at a slightly higher pH than your stomach (pH closer to neutral than the pH of 2-3 in your stomach).
The rumen does not really have protection against acids like your stomach does. And in fact, in certain situations, if the ruminants eat food (corn, usually) that is fermented by acid-producing bacteria, the pH drops too low and causes real severe problems (rumenitis, lactic acidosis, etc.).
Keep in mind, again, that ruminants have multiple stomachs to digest their food. The cud comes from the rumen (and maybe reticulum). The stuff that goes to the abomasum, their equivalent to our stomach, is not supposed/meant to be refluxed.
Short answer: Their cud is more basic and less acidic than what you reflux, and is meant to be that way, and it doesn’t harm the animal.