This page of our Ghost of Tsushima guide contains the answer to the question of how to change the saddle placed on the player's horse. There are many different types of saddles to choose from and some of them are quite original in appearance.
One important note to start with - saddles are cosmetic items, so they have no effect on the gameplay and do not guarantee any new passive skills. Switching the saddle does not affect the statistics or the way the horse travels.
The main requirement to unlock saddle switching is completing Hammer and Forge story quest, which involves capturing Komatsu Forge. After completing the quest, a special NPC will appear that sells new saddles.
Saddles are not "bought" for supplies. Instead, you have to look for Sashimono Banners, which are one of the categories of collectibles in the game The banners are mainly found in large locations occupied by the Mongols. However, finding them is sometimes one of the mandatory objectives for taking over a given location.
While collecting banners, wait for information about the possibility of picking up a new saddle. It will appear every time you collect a certain number of banners from the 80 scattered around the game world. The first such announcement will be made after 5 flags have been collected.
Upon reaching Komatsu's forge, meet the monk to automatically hand over the collected banners to him. You will be able to select a new saddle from the list and it will automatically be placed on the hero's horse.
You can manage your saddles from the equipment tab level. There's no problem going back to a previously used saddle.