Movement vs expansion in space

How can an observer distinguish two objects that are moving from each other in space compared to those that are moving with space due to expansion of the universe?

They could compare each observer’s motion to the local inertial frame where the cosmic microwave background radiation is isotropic.

Dale said:

They could compare each observer’s motion to the local inertial frame where the cosmic microwave background radiation is isotropic.

How do We know that CMB is not moving itself, because the content of the universe was moving during decoupling?

exander said:

How do We know that CMB is not moving itself, because the content of the universe was moving during decoupling?

Because we arbitrarily choose the reference frame where it is isotropic. We do that because it makes the math easier and we arbitrarily prefer easy math over difficult math.

Not sure if your question is how we can tell two objects are comoving or how we can tell the situation of expansion apart from objects just moving away from each other. Dale has answered the first of those question. The answer to the second is that, locally, you don’t. See https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

exander said:

How do We know that CMB is not moving itself

Because seeing the CMB as isotropic is the meaning of "moving with space due to the expansion of the universe", which is what you asked about.

exander said:

How can an observer distinguish two objects that are moving from each other in space compared to those that are moving with space due to expansion of the universe?

The basic point is that it is spacetime that underlies everything. There is no "space" or "time", just spacetime. If you want to talk about space or time separately you have to first slice 4d spacetime into a stack of 3d slices, each of which is "the universe at one time". But there isn't a unique way to do this. There are no gridlines built in to spacetime telling you how to slice (although there are limits about which direction you can sensibly call time).

So you cannot tell if you are moving in space until you define what you have chosen to mean by "space". In cosmology it turns out that the slicing where the CMB is the same temperature at one time and isotropic when you are stationary is so useful that nobody uses anything else. But the underlying physics doesn't care about your mathematical convenience. You could use another slicing. There is no physical measurement that can tell you what choice you made.

exander said:

How do We know that CMB is not moving itself, because the content of the universe was moving during decoupling?

We know that the CMB has no net motion in space because we defined space to be the slicing of spacetime where the CMB has no net motion. It's true by definition.

A non-tautological question would be: is the FLRW spacetime an accurate model of the real spacetime? Or, equivalently, are there observations which are inconsistent with us being in an FLRW spacetime (or at least one that looks FLRW on large scales)? The answer to those is a resounding maybe.