Like said in the comments, and also found in the RDR2 Companion app, you can saddle a horse by pressing L2 to target the horse, and d-pad up te remove and place saddles.
This will be available after the turtorial, in chapter 2. When doing the 'Exit Pursued by a Bruised Ego' mission given by 'Hosea Matthews'. In this mission, the horse and saddle mechanics are explained.
You Can buy and sell Horses at stabled throughout the world. You can break wild horses, or steal owned horses. You can ride a horse temporarily, but to make it yours, you'll need to put your saddle on it.
To saddle a Horse, approach it with a saddle in hand and hold D-pad up to put it on. You can only have one saddle and therefore one main Horse, at a time. To remove a saddle you must approach and hold L2 to lock on and hold D-pad up to remove. You can also saddle a horse by taking it to a Horse Shop, at this point your previous horse will be added to the stables.
Arthur cannot saddle, bond with, or own Horses belonging to other members of Dutch's gang.
Remove your saddle from a dead horse by approaching it and holding D-pad up to remove. Saddles left on dead horses can also be recalled using the stables.