Generally what
@figbat said already. Saddles - other than sidesaddles - aren't 'handed' (with an exception as mentoned below) and sidesaddles can be/were made for the 'other' side too so that people (ladies or others with lower limb/hip issues) could ride on the 'other' side if necessary.
A horse trained for 'normal' sidesaddle would then need some degree of training/accustomisation for the 'other' side.
Horses are as 'handed' as any other animal, including ourselves, and of course strongly influenced by the handedness of their early (and continuing) trainers and handlers.
A horse might be a bit puzzled or upset - depending on its general level of 'quietness', trust, desensitisation or whatever you want to think of it as - by being handled, saddled, mounted etc from what is often thought of as the 'wrong' side. I've always trained my horses to accept stuff - at least occasionally - from both sides although it can be
really really difficult for the human to do some things from the 'wrong' side! I'm fairly ambidextrous so things aren't as difficult for me as they would be for many people, even so, having learnt to mount from the left hand side of the horse since I was about two years old, I never did master conventional mounting (foot in stirrup etc) from the right hand side, although vaulting on in my gymkhana days, and in my more mature years climbing a gate, a wall or a mounting block, to get on from whichever side was most convenient, was a doddle.
A really cheap 'felt saddle' such as might be used on a donkey or a quiet pony would often have the adjustable girth buckle only on the left-hand side; the other side would be sewn on.
A conventional bridle will usually have adjustment buckles on both sides for the cheek pieces, but the throatlatch will have only one buckle - on the left hand side - and most nosebands will also have their buckles only on the left hand side - one for adjusting the length/height of the noseband and one for adjusting its tightness round the horse's 'face'. HOWEVER the fastenings on a harness horse's bridle are different to those on a riding horse' bridle and there are more points of adjustment, usually all pieces are adjustable on both sides.
There are many more styles than the 'conventional' nowadays, though, and Western-style bridles are very different, sometimes with an earpiece serving to replace both throatlatch and browband, and adjustments of different types used, and both riding tack and driving harness made of many of the synthetic materials now available is sometimes different in style and fitting details.