The following presumes a system at uniform temperature and pressure throughout.
What do we mean when we say "uniform composition"?
We mean that the composition averaged over a certain spatial volume of the substance remains the same moving through any location in the substance. The spatial volume may be the size of atom or molecule in a pure substance. It may be the size of a sufficient volume to contain a representative sampling of all chemical constituents in the substance as a mixture.
Air can be assumed as a pure substance because no matter where I take a unit volume, the chemical composition - the percentages of its constituents (nitrogen, oxygen, CO2 etc) will remain same everywhere.
Does calling a material homogenous the same as calling it a pure substance?
No. Air can be assumed to be uniform in composition over a spatial volume that is about the size on the order of magnitude of the mean free paths of the molecules in motion in the gas. A substance that is uniform in composition is not always a pure substance (e.g. air), but a pure substance is by default always uniform in composition (e.g. water).
What is meant by "same properties in a direction"?
First off, a materials' property is a moderating factor between an external stimulus and the response of the substance. By comparison, an inherent property of the substance is one that exists simply because the substance exists. Density is an inherent property. Transparency is a materials' property because we have to shine light on the substance to obtain its value.
We do not associate inherent properties with the need to measure in a specific direction in the substance. Instead, we say that in a homogeneous substance, inherent properties are always the same in all locations.
The additional statement applies to materials' properties. When we want a materials property, we have to apply an external stimulus. We often neglect that we do so in a certain direction relative to some fundamental arrangement of the chemical constituents in the material. When we shine light into a single crystal of quartz, the transmission will depend on which crystallographic direction we shine the light. Why? Because the atomic arrangements in quartz are different in different crystallographic directions. This anisotropic behavior of transmission is less apparent in a polymer. Why? Because the random arrangement of polymer molecules has no preferred direction when viewed externally. So the transmission of light is isotropic, meaning it is the same regardless of the incident direction of the light beam onto the material.