Only the highest enhancement bonus applies
The Dragon #318 article "Oriental Adventures Update: Eastern Flavor" on Magic Chahar-ainas and Dastanas says
Both dastanas and chahar-ainas provide special armor bonuses to AC that stack with other armor bonuses granted by certain forms of light armor. However, it is still the case that only one enhancement bonus can apply to a character’s armor bonus at any time. This, if a character wears +2 dastanas, a +1 chahar-aina, and +1 cloth armor, only the +2 bonus fom the dastanas increases his AC. A character can still gain the benefit of special abilities attached to multiple pieces of armor, however, so a character wearing +2 balance dastanas and a +2 displacement chahar-aina has a +2 enhancement bonus to AC and can use the balance and displacement abilities. (42)
Note that the Arms and Equipment Guide represents the dastanas (15), superseding Oriental Adventures, but changes little except putting the dastana into the Shield category exclusively (rather than OA's Shields and Other Additions category). The chahar-aina remains exclusive to Oriental Adventures.
There are magic and non-magic items made from adamantine
While the DMG lists Adamantine Armor as a magic item, some items made from adamantine are not magic.
Gargoyles resist damage from “nonmagical weapons that aren’t adamantine” (MM, p 140). It follows there would be nonmagical weapons that are adamantine.
So, adamantine weapons would strike gargoyles similarly to how magic weapons would, without being magic weapons, or even magic items. Mechanically, they are similar to silvered weapons (the phrasing in monsters’ resistances blocks are the same) but they defeat a different set of monsters’ damage resistances.
Xanathar’s Guide does not call Adamantium “Magical”
Xanathar’s Guide to Everything (p. 78) has a section on adamantium weapons which describes adamatine as “an ultrahard metal found in meteorites and extraordinary mineral veins” but does not mention it being magical.
Xanathar’s lists the properties of weapons “made from or coated by” adamantine. These are distinct from the proprties of magic items. For example, adamantine weapons don’t get magic item resiliency.
Official Example (Spoiler Alert)
There is an example of such a weapon in official Wizard’s 5e material.
In The Lost Mine of Phandelver, the Spider Staff is a “black, adamantine staff” that “can be wielded as a quarterstaff.”
Note, the staff does not strike as a “magic quarterstaff” like the Staff of Power does. It is a magic item but not a magic weapon. As far as whomping things goes, it is a “nonmagical weapon made of adamantine.”
For mithral items, the rules say very little
I’m not aware of any mention of mithral in the core rules outside of the magic armor. A DM might follow the example of adamantine — that it’s a rare metal that is often found in magic items. Or they might rule it is inherently magical.
Magic Items are defined as such. Custom items are defined by your DM.
If an item is listed as a magic item then in general they would have the qualities common to magic items of their type. A custom item invented by your DM might have any properties.
The particular qualities of any particular item in your game are of course up your DM. Your character might (and probably should) learn whether an item is magical or not, and its properties, resistances, etc., by means such as an Identify spell.
Best Answer
TL;DR. Saying that enchanted armor breaks bounded accuracy is somewhat incorrect, but excessively high AC does break bounded accuracy and limiting the availability of enchanted armor is the most convenient solution.
Excessively high AC breaks bounded accuracy
To understand why, you need to understand that bounded accuracy exists to achieve a list of desirable goals. Breaking bounded accuracy doesn't necessarily mean that the game becomes unplayable, it just means one or more of those desirable goals has not been met. However, a lot of DM resources are written with the assumption that bounded accuracy is respected, hence when it is broken some of those resources become unreliable or incorrect.
The article "Bounded Accuracy" on Legends and Lore (an official D&D website) by Rodney Thompson (who, at the time, was a WOTC employee and is credited with Rules Development on the 5e PHB, among credit for other WOTC products) does a great job of explaining what those desirable goals are.
A few of those goals are relevant to this question. Among them, the most relevant is the following:
In other words, the monster's usefulness should depend mostly on their damage, not on their bonus to hit, so that the DM's roster (the list of useful monsters) never shrinks even as the party becomes more powerful.
A monsters that only has a 5% chance to hit, is not useful. The DM would have to run an impractically large number of said monsters for them to have a significant impact on a combat. Even monsters with a 15% chance to hit are borderline impractical: there's a very real chance such a monster will never hit over the course of several turns and that its only impact on the combat is to waste time.
Imagine that half of the party has an excessively high AC, for example 26 AC. In this situation, the DM's roster shrinks to exclude monsters with +6 bonus to hit or lower and even monsters with a +8 bonus to hit are borderline impractical. In other words, the desirable goal has not been met and bounded accuracy has been broken. It just so happens that +3 full plate plus +3 shield is 26 AC.
Excessively high AC is desirable from a player's perspective
There are a lot of ways to increase AC and most of them stack:
If a majority of the those things are easily available, then it's possible for PCs to have excessively high AC on a regular basis.
And it's not just possible, it is also desirable from the player's perspective, because naturally the player doesn't want their character to be hit. Moreover, increasing AC even yields increasing returns, for example against a +9 to hit: increasing AC from 15 to 16, decreases damage by 1/15; increasing AC from 25 to 26, decreases damage by 1/5.
This is a problem, because it means the game provides both the means and the incentives to break bounded accuracy in the context of AC.
Limiting magic items is the most convenient solution
Of all the things which increase AC, magic items are the only ones entirely under the DM's control, hence the easiest way to preserve the AC's bounded accuracy is to limit the availability of those magic items.