The Russian Jagers were by no means the only troops to carry this, and you have essentially answered your own question: the "blanket-like sash" is simply that -a blanket. This provided each individual soldier with a means for shielding himself against the environment, not only by providing a rudimentary bedroll for when better quarters were unavailable, but also for daytime as something to sit on during breaks and mealtimes, as a rain poncho or windbreak and so on.

Essentially a warm, preferably somewhat water resistant woolly tarp/blanket with the natural wool fat intact was (and is) an essential part of outdoors equipment even though the exact material changes over time.