I don't really understand what we are talking about when we say space is homogeneous. What we are measuring? My notion is: it should depend on the entity and with respect to that entity one can decide space is homogeneous or not! May be I'm asking stupid question, actually the fact is I don't understand the phrase "space is homogeneous" or "space-time is isotropic"
Suppose we measure the density of matter as a function of position. Then we will come up with some function for the density $\rho(r, \theta, \phi)$ where $r$, $\theta$ and $\phi$ are polar coordinates with ourselves at the origin.
The density is isotropic if it is the same in all directions i.e. it is independent of $\theta$ and $\phi$ and just a function of distance, $\rho(r)$.
The density is homogeneous if we can move our origin, i.e. move to any other point in the universe and our density function $\rho(r)$ is unchanged.
The simplest such function is if $\rho$ is just a constant since it then automatically the same in all directions and the same at every point.
I've used density as an example, but this applies to any property of the universe that is a function of position.
Our universe is obviously not isotropic and homogeneous because in same places there is a vacuum while in others there are stars. However we know it started out as very close to isotropic and homogeneous from measurements of the cosmic microwave background radiation. And if we take average density on a large enough scale then we believe it is still on average isotropic and homogeneous (though this is a somewhat controversial issue).
You are correct in the sense that there are physical quantities, observables, like scalar and vector fields that are, in general, varying in space.
However, that's not what the phrase "spacetime is isotropic" is meant to convey. What this means is that, in the absence of anything to break the symmetry, each point in spacetime is no different than any other point in spacetime.
Without reference to anything else, spacetime is considered as homogenous everywhere.