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.