The atlas already holds astragaloi, the ankle bones thrown for divination and games in the Greek and Roman world, and the resemblance to shagai is exact rather than approximate. The same bone, from the same part of the same kind of animal, with the same four landing faces, thrown in the same small numbers, read for answers and played with for entertainment, across both cultures.
It is tempting to make that into a transmission argument, and it does not support one. A sheep has two astragali and every pastoral society kills sheep. The bone is nearly cubical, resists rolling, lands reliably on one of four faces, and is produced in quantity at no cost. Anywhere people keep sheep and want a randomising object, this is the obvious one, and it would be surprising if it had not been found independently many times over.
What comparison can honestly do here is narrower and more useful. It can ask which faces are given which values, what the combinations are called, and how the reading is structured, because those are cultural rather than anatomical. The Mongolian naming of the faces after four herd animals is specific to a place where those four animals are the herd, and that is the level at which the object tells you something about the people using it.