Link can obtain other bridles as rewards in mini-games. The Traveler's Bridle can be obtained by using the amiibo Rune with the Link Rider amiibo of the Breath of the Wild series. The Monster Bridle can be purchased at the Fang and Bone using the shop's special Mon currency which can be obtained by sell monster parts to the owner Kilton. Link can change bridles at certain stables by talking to a female stable hand. However he must have fully bonded with a registered horse in order to change its bridle and must have obtained a different bridle from the initial set first.
However some horses such as the Giant Horse from the side quest "Hunt for the Giant Horse" or Epona, obtainable through the amiibo Rune, are equipped with special bridles that is unique to them and cannot be changed. However the stable hand will state that Epona will not like it. In the case of the Giant Horse, his size prevents Link from changing bridles as only its default one fits.
Unlike armor worn by Link, bridles are purely cosmetic, and do not affect a horse's stats or increase their defense. The sole exception is the Ancient Bridle (DLC), which increases a horse's stamina meter by two spurs. (It is also claimed that the Knight's Bridle reduces damage, but this requires testing.)