It does mean that indeed! Note for example that in a gravity field like the one near the Earth surface the "down" direction has very different properties than

It does mean that indeed! Note for example that in a gravity field like the one near the Earth surface the "down" direction has very different properties than the two horizontal directions (1). In particular, an object stationary with respect to the surface released at altitude will develop linear momentum in the down direction, so in the down direction momentum is not conserved. OTOH, the linear momentum in any of the horizontal directions is conserved. (1) Isotropy means that space has the same properties in all directions.