This answer I'm about to offer comes from my grandmother As I recall, the many times I spent with her. She was born in 1906. Having her mother die 1909, she was raised by her grandparents. The "container" hanging below the axle was for multiple purposes. Yes axle grease may have been there. More commonly put there, as I have been told, was embers from the night or morning camp/cook fire. it was the safer place to put such an item. not having lighters,matches readily available. coals and embers could be stored there fires were used for cauterizing wounds. and the time spent waiting for one could have cost the life of a loved one,,the family milk cow,or a good mule. (grandma loved her mules!) axle grease mostly obtained from animal fat was wrapped in large hide and the hide itself was custom cut to fit as needed as bearings were more of a "direction" than mechanical equipment. hope this info. helps. thanks for asking!
How lucky you are to have these stories from your grandma write them down so they can be passed down in your family lucky you my grandma use to work for wild bill in North Platte Nebraska and she would always tell us stories when we would visit her