Can the reference frame be "inertial" or "non-inertial"?
Inertia is a property of matter, and the reference frame is an abstraction. Is it possible to make such contradictory assumptions in speculative modeling as awarding the property of" inertia "to the abstract concept of "reference system"? Isn't this the reason for the inconsistencies of theories from different branches of physics?
First, the inconsistency of theories in physics is a bit of an exaggeration.
Secondly, physics (and science in general) cannot be judged by the sound of terms, “linguistics”. The term has a specific definition. In particular, an inertial frame of reference is one in which the laws of Newtonian mechanics are fulfilled (for small velocities).
Yes, this is an abstraction, a way of describing it. No deep conclusions should be drawn from the consonance with the word “inertia”. And you will not find the concept of “inertia” in modern physics.