This design is a custom holder for this Medical Kind brand walking cane, though a step version is included to facilitate potential adaptation to other models. I tested it with a different cane and other than it being snug because of its slightly larger diameter it worked pretty well, so only a very minor change to the model would be required. I suspect that means it ought to be possible with other canes as well.

It exploits the slightly larger diameter of a small metal collar that locks the handle to the cane (a feature common to many canes), allowing it to be lowered into a socket that holds it. To ease insertion, angled arms are included and the socket is tapered. To remove, just lift up a small amount and slide it out sideways.

Mine is mounted under the lip of a shoe cabinet, which prevents it from rotating around the single screw, so other placements may require another fastener to be added to the model.

I had good luck printing this in PETG with no supports. I suggest using a fair number of walls for the narrow arms to be strong.