Yes, but not the mithral armor underneath it.
Mithral (and adamantine) armors are listed under the "Magic Items" section of text in the DMG, and are listed directly in the magic items tables. They follow the rules for magical items, and are treated as magical items for any terms or conditions that discriminate based on such.
JC confirms in this twitter post circa 2017 when it still meant something.
However, if the default ruling at your table is that mithral and adamantine don't count as magical items, a leather tunic could be considered clothes with an AC value of 10. Another attack from the black ooze reads
Pseudopod. Melee Weapon Attack: +5 to hit, reach 5 ft., one target.
Hit: 6 (1d6 + 3) bludgeoning damage plus 18 (4d8) acid damage. In
addition, nonmagical armor worn by the target is partly dissolved and
takes a permanent and cumulative −1 penalty to the AC it offers. The
armor is destroyed if the penalty reduces its AC to 10.
In this reading, the tunic would be destroyed on first contact unless it's magical. But this reading also implies that none of my assumptions about the rules in your game are necessarily accurate, so you might want to have a talk with your DM.