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)
Armor Proficiency. Anyone can put on a suit of armor or strap a shield
to an arm. Only those proficient in the armor’s use know how to wear
it effectively, however. Your class gives you proficiency with certain
types of armor. If you wear armor that you lack proficiency with, you
have disadvantage on any ability check, saving throw, or attack roll
that involves Strength or Dexterity, and you can’t cast spells.
Summed up: Anyone wearing armor must be proficient, or face the consequences.
PHB, Chapter 5, page 155: (on barding as equipment, price)
Barding. Barding is armor designed to protect an animal’s head, neck,
chest, and body. Any type of armor shown on the Armor table in
this chapter can be purchased as barding. The cost is four times the
equivalent armor made for humanoids, and it weighs twice as much.
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)
Variant: Warhorse Armor
An armored warhorse has an Armor Class based on the type of barding worn
(see chapter 5 for more information on barding). Its Armor Class includes the
horse’s Dexterity modifier, where applicable.
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.
It was in the very first incarnation of D&D. Witness ye, the words of OD&D (Men & Magic) from 1974:
Magic-Users: Top level magic-users are perhaps the most powerful characters in
the game, but it is a long, hard road to the top, and to begin with they are weak,
so survival is often the question, unless fighters protect the low-level magical types
until they have worked up. The whole plethora of enchanted items lies at the
magic-users beck and call, save the arms and armor of the fighters (see, however,
Elves); Magic-Users may arm themselves with daggers only.
And therein lies your answer. A core part of D&D balance from the beginning has been that wizards wield incredible power but are fragile physically and can't use armor or most weapons. Gygax made that a part of the game, as it was how he envisioned his fantasy world grafted to a wargame to work.
The in-fiction justification back in the day for this restriction was simply that 'that class doesn't get around to learning that." Proficiencies as a formal idea that you could take instead of just having a monolithic bundle of abilities based on your class didn't come till later. If you dual or multi-classed, you could cast wizard spells in armor just fine by the way, there was originally no real inhibitor except that "wizards don't learn that in wizard school."
However, even "proficiency" is a compelling argument - armor's not "just clothing." Untrained people put on wetsuits, climbing harnesses, etc. in laughable, inefficient, binding, and frankly dangerous ways. Football players spend a lot of time micromanaging their pads and helmets and learning to move in them. The idea that "I'll just slap this armor on it'll be fine" falls down when its specific adjustment is what keeps you from getting bones broken from deflected blows, or from it getting caught on the battlefield/foes/weapons and dragging you to your doom.
One can also argue the influence of genre tropes (Gandalf didn't wear armor!) on this long-standing trope, but that's pretty much an opinion-fest, and is already on this SE as a closed question: Where does the stereotype that wizards can't wear armor come from?
Armor Across The Editions. According To The PHBs
0e, 1e, and 2e: Magic-users couldn't use armor because they weren't trained in its use, period. They are busy learning spells from books instead, and armor is a bit binding and impedes somatic components. Races that could multiclass or dual classing in general let you cast magic-user spells in armor.
3e, 3.5e, and Pathfinder: Magic-users can gain proficiency in armor but even then there's a spell failure chance for spells with somatic components because of armor's restrictive nature.
4e, 5e: Armor has no specific effect on spellcasting, though if you're not proficient you take various penalties to everything including spellcasting.
As you can see, the approach has really been quite consistent. Even before there were proficiencies, and after, the general explanation is "if you aren't proficient in armor, then you will have trouble with your spells," though that penalty has lessened over the years. It's a mix of game balance and realism - the same reason a wizard doesn't know armor and weapons is the reason a warrior doesn't know spells - in life, you have to make choices about what you learn, and "all of it" is not a feasible answer, at least not as a 16-year-old starting adventurer! In earlier editions it was harder to learn things in general as it was very class-based; now that there are proficiencies and stuff a wizard can learn armor like anyone else, by making that tradeoff to not learn something else useful.
Best Answer
He's in a booby trapped iron safe. He's safe. He's also trapped.
His situation has a number of upsides: He's in an iron safe. The same walls of iron that kept him out, keep everyone else out. If people can attack him through the safe, it's not particularly safe, is it?
His situation has a number of downsides: He's in an iron safe, The enemies don't need to attack him, they just need to keep him there. He certainly won't have enough food, water, or sanitary facilities to last out a siege, if they're feeling patient. If they're not feeling patient, they can simply open it, with all due preparation. If they're feeling sadistic, they can pile firewood under it and roast him alive.
Your player is safe from melee attacks in the safe. He is not actually safe.