The universe is not isotropic in the time dimension. However, across space the prevailing thinking is that the universe is isotropic on a large scale in the spatial dimensions.
As far as I understand (I'm not a physicist), to be able to say that the universe is isotropic across space, you need a point of view that just doesn't exist. You cannot speak about an absolute synchronicity among events happening in locations scattered among an arbitrary subset of the set of all visible galaxies, and without that, you cannot abstract time out of the picture, which you'd need to do to speak about isotropicity across space.
I don't think that, in this regard, you can say much more than "the universe is not isotropic across space-time".
You cannot speak about an absolute synchronicity among events happening in locations scattered among an arbitrary subset of the set of all visible galaxies
The cosmic microwave background gives you a physical realization of just that (but of course only approximately so), at least as far as cosmologists are concerned. The rules of relativity of course still apply, making this particular synchronicity convention just one of many others...
But the cosmic microwave background refers to one single event, that has echoes everywhere, right? If I understand correctly, you don't need to synchronize anything to have echoes everywhere.
But the cosmic microwave background refers to one single event, that has echoes everywhere, right?
An event is a single point in spacetime, whereas photon decoupling happens everywhere, defining a spacelike hypersurface we use for synchronization (in the idealized scenario).
Subsequently, the CMB allows us to single out a particular reference frame (the one where it looks isotropic) and provides a measure of expansion via its redshift/temperature which we can then translate to cosmological time (ie time since the big bang as measured by an observer following the Hubble flow) via our cosmological models.