Just tried an experiment. Made a new world in creative, made a bunch of donkeys and mules with spawn eggs. Could put chests on donkeys and mules. Changed to Survival, could also still put chests on donkeys and mules.
Also tried breeding a horse and donkey in survival mode (parents were tamed in creative mode) and could tame and put chests on it.
Perhaps it is a bug in wild spawned donkeys and mules vs ones made with spawn eggs? So far I can only reproduce on a pure survival world.
Also having the same issue, both on Windows 10 and Xbox (Bedrock) 1.2.13
Pure survival world on Realms. Never been set to creative (I earn achievements)
Tamed donkey, bred a and tamed mule, cannot put a chest on either. Chests on Llamas works okay.
When I logged back in to my realm yesterday on the Xbox, all maps were operating normally. I will try to see if I can reproduce on a new world that I can attach. This particular world is 53 megs right now.