When we say rotation relative to the observable universe we mean more exactly relative to the bulk mass-energy of observable universe.

Any reference frame rotating relative to the observable universe is commonly called just a rotating frame (or rotating reference frame). Rotating frames are NOT inertial frames (i.e., they are non-inertial frames) In fact, every point in rotating frame has its own acceleration, and so the conversion of the whole rotating frame to an inertial frame (see subsection Non-Inertial Frames Converted to Inertial Frames) requires a special treatment which we discuss below in section Rotating Frames and the Centrifugal Force and the Coriolis Force.

Since being in rotation relative to the observable universe creates a non-inertial frames, being in rotation relative to the observable universe is in a sense absolute rotation. However, since Wikipedia: Absolute rotation using the term absolute rotation in a somewhat different way, yours truly prefers to say rotation relative to the observable universe rather than absolute rotation.

How do we know that the theory is true that rotation relative to the observable universe creates a non-inertial frames?

A combination of general relativity and current cosmological models tell us the theory is true and so far all observations are consistent with the theory.

We can actually measure rotation relative to the observable universe to high accuracy/precision by measuring rotation relative to cosmological remote astronomical objects (see Wikipedia: International Celestial Reference System and its realizations) and such measurements are done when we want to measure rotation relative to the observable universe to the highest accuracy/precision.

Note all reference frames including inertial frames NOT rotating relative to the observable universe are all NOT rotating with respect to each other.

Actually, a qualification is needed in that there may be reference frames that are inertial frames (in a sense) rotating relative to the observable universe in very strong gravitational fields such as near black holes. But yours truly CANNOT find any reference that elucidates this qualification. It is hinted at by Wikipedia: Inertial frame of reference: General relativity. Yours truly will usually NOT refer to the qualification again.

Note the qualification is required by general relativity and is NOT present in pure Newtonian physics as it is NOW understood with free-fall frames being the elementary inertial frames.

below UNDER RECONSTRUCTION BELOW

Comoving frames have origins that participate in the mean expansion of the universe.

In fact, every point in the observable universe defines the origin of a comoving frame. and so comoving frames are everywhere and you can use thos comoving framese for calculations in principle, but in practice there are limitiations.

Explication:

1. General relativity plus various simplyfing cosmological assumptions give the Friedmann equation, the most basic equation of cosmology.

   Solutions to the Friedmann equation (i.e., Friedmann equation models) dictate that the observable universe should be in general expansion: a literal growth of space according to general relativity.

   If the mass-energy of the observable universe were spread out uniformly throughout the observable universe in what is called a perfect fluid, there would just be a general scaling up of the distances between all point masses.

   The expansion of the universe begins as an initial condition of the Friedmann equation models. The universal gravitational field tries to decelerate the expansion and the cosmological constant field (or its dark energy equivalent) tries to accelerate the expansion.

   In the Friedmann equation model the Λ-CDM model (standard cosmological model c.1995--) (see Douglas Scot, 2018, The Standard Model of Cosmology: A Skeptic's Guide, p. 10), cosmological constant field is winning and there is currently acceleration of the universe.

2. Now if the mass-energy of the were spread out in the form of perfect fluid as described above, every point in the observable universe would define a free-fall frame (i.e., an elementary inertial frame) in the LOCAL combined gravitational field and cosmological constant field.

   A test particle at any point would then exactly participate in the mean expansion of the universe: i.e., it would stay at rest in its LOCAL free-fall frame.

   Those all points would comove with the expanding universe and so all points would define a LOCAL comoving frame.

3. However, the perfect fluid idea is just one the simplifying assumptions used in cosmology used to derive the Friedmann equation. The expansion of the universe dictated by the Friedmann equation is just an average behavior for the observable universe.

   Really, of course, the mass-energy is clumped into stars, galaxies, dark matter halos, non-uniform intergalactic medium (IGM), etc.