It's called Armor.
What do the Rules say? I think the applicable rules are armor rules. The intent behind wrapping chains around yourselves is protect yourselves. That is the job of armor.
Chain fashioned around your self are similar (enough) to Chain mail or ring mail, both are heavy armor. Chain mail requires strength of 13 and imposing disadvantage to stealth. For that you get an AC of 16. Those chains have no HP, and no mechanism for it to be destroyed for simplification reasons.
Adding other objects to the character to act as armor isn't necessarily outside the rules, but isn't something the rules consider. Which means that what happens is left up to the DM. The most appropriate rules for this sort of thing are the armor rules, not the object HP rules. Adding armor in this fashion should affect AC, not add additional HP, and depending on the amount of chain likely have stealth implications.
Object Rules
The barrel is what item HP are meant for, a situations that happens to come up where someone needs to destroy an object. The rules on objects are for objects, not armor. The rules for objects can be found in the SRD ( official ) in Game Mechanics under Objects. Reading them it is clear that the intent is an item caught in the cross fire, or the characters try to destroy an object. It isn't meant for items that are worn to provide temporary hit points.
This path might inspire your DM to employ armor/weapon damage house rules. Your swords might suddenly break or shields shatter. That is the logical extension of applying these HP rules to all things in the world.
Ultimately, it's the DM's Call
To apply armor rules or object rules is the DMs choice. Armor rules are the ones intended for defense, but there is nothing preventing him from applying object rules the way he has already.
You have to take the Moderately Armored feat to get Shield proficiency
It doesn’t matter that you have Medium Armor proficiency from your race; proficiency with Shields is a different proficiency and the Moderately Armored feat is the only way to get it without multiclassing.
This is not an oversight. The designers wrote it that way in plain English so that was what they intended to write.
Best Answer
If it's made by dwarves, we call it dwarven. If it's made by elves, we call it elven. Just because it's called Dwarven Plate does not mean it's made only for dwarves.
For magic items (which is what Dwarven Plate is), page 140 of the DMG states;
And as Quentin states, for general rules on armour there's page 144 of the PHB;
There's also nothing in the item's description that restricts it to only dwarven characters. Unlike the Dwarven Thrower which specifically says it needs to be attuned to a dwarf.
So, as a general rule, whether found or custom made, (within reason) it can be worn by any race.
It's then up to you to decide what limitations/realism you add to it I.E if it's found, does it auto adjust to fit non-dwarves or would they have to find someone to adjust it for them, are the dwarves actually willing to make this very rare magical armour for other races without good reason.
One thing to remember though is be consistent. If you put restrictions on one piece of armour (magical or not) you should also apply restrictions to others.
Another thing to consider is to ask the other players (and DM if you're not DMing) what level of immersion they want.
Do they want to simply be able to wear any armour they come across or do they want that extra bit of realism that makes them drag their loot half way across the country to find someone willing the make the necessary adjustments.