According to my understanding, it is a Roman adaptation of the Epitrachelion and Orarion - essentially priestly neck-garments. I wouldn't read too much into the clothing style itself, as it changes infrequently and often erratically and illogically. But the garment itself, functionally, is simply the symbol of the priesthood or diaconate, but styled differently. It's also important to note how simple they can be, so it is likely that they were adapted as priestly garments some time after they were in common use by aristocracy.
The interesting part is that while in the Orthodox tradition the stole is two different garments for the priest and deacon, in the West it is almost the same garment simply worn differently, albeit with the final effect being identical: the priest's is symmetrical and comes down like a beard, and the deacon's is asymmetrical and worn like a sash.
This suggests a common origin to all of the stoles, but does not tell us what the proto-garment was, or if one of the two was the proto-garment. I would be inclined to assume the Orthodox garments, partly because of the Oriental Churches (Non-Chalcedonians):
etc... the naming is Western (stole) but it seems like the garment is mostly unchanged and similar to the Eastern Orthodox one.