I'm playing a level 11 monk/barb in D&D 5e. (I've put 8 levels into monk and 3 into barbarian.)
I recently got the Eldritch Claw Tattoo from Tasha's Cauldron of Everything (p. 126), and I also have the Insignia of Claws (ToD, p. 178; HotDQ, p. 94). These magic items both add a +1 bonus to the attack and damage rolls of my unarmed strikes.
Would this allow me to add a +2 to my the attack and damage rolls of my unarmed strikes?
Yes, these stack.
The rules for combining game effects on p. 252 of the DMG - added in the DMG errata - state (emphasis mine):
Since Eldritch Claw Tattoo and the Insignia of Claws are different magic items, they can affect you at the same time. The restriction on stacking applies to features with the same name.