States that all matter is evenly distributed throughout the Universe, in the same amounts, moving in the same direction and in the same ways

This is the principle of homogeneity. While on a small scale, we have that things are very different (we have wood, houses, jam etc.). But on a galactic scale, even beyond the scale of galaxies, matter is distributed in the same way (same density, same movement etc.). This yields that assuming that matter has everywhere the same density and movement is a good approximation to our Universe. This is what the principle of homogeneity encompasses; it is an axiom that states the premise of the question.