Mounts can be given barding to duplicate the effect of armour at 4x the cost. However, barding makes no statement about armour proficiency, nor does the stat block for some of the more obvious mounts (Riding Horse, Warhorse, etc.).
If a mount is given barding, is it proficient in it? If not, how is it possible for them to gain proficiency?
The penalties for wearing armour without proficiency are no spellcasting and disadvantage on ability checks, saving throws, and attacks that use Strength or Dexterity.
Assuming they are not proficient, do mounts with barding have disadvantage on Strength and Dexterity -based ability checks and saving throws?
And finally, non-humanoid creatures are not explicitly stated to use either Strength or Dexterity in their attacks, but I would assume physical attacks have to use one of these two, so:
Assuming they are not proficient, do mounts with barding have disadvantage on attacks?
As far as I can tell, rules are scarce on this subject. All we have is:
PHB, Chapter 5, page 144: (on wearing armor, in general)
Summed up: Anyone wearing armor must be proficient, or face the consequences.
PHB, Chapter 5, page 155: (on barding as equipment, price)
Summed up: You can put armor on an animal, and it's price is four times that of normal armor.
PHB, Appendix D, page 310: (variant rule on warhorse specifically)
Summed up: A warhorse can wear armor. Warhorse, as opposed to Riding horse. Since this is a variant rule, it implies that without the variant rule in effect, all animals (not only warhorses - and war-dogs, war-elephants etc too, I presume) can wear barding.
But do they require proficiency? As per the general rule on armor proficiency quoted at the top; yes. How would an animal go about acquiring such a proficiency? We're not told. Also, we're not told what the difference is between a riding horse and a warhorse, and if/how the first can become the latter.
However, when the RAW is lacking, we just keep rolling. The spirit of the game has always been "make it up as you go". So here is what I would suggest: A non-war-trained mount is not proficient with armor. A war-trained mount is. I believe this is relatively consistent with the rules above.
Also, we need to come up some rules for training a war-animal. Say a month or two of rigorous training in the hands of a skilled person?
Yeah, and just so I've mentioned it: A lawyer might argue that barding and armor are two different things. I'd argue that it's the same thing with different names, depending on the wearer.