Mule deers require 2.xx Taming skill - I think some of the info is missing here. Tested it with 3 different mule deers, even a legendary one, please see screenshots:
I've tamed an animal and put it in the stable, so I have no active pet. When trying to tame a second pet, I get the message that I already have the maximum number of pets. Is this due to my low taming skill? Please see screenshot.
Just an info, no action needed: Mutants now have definitely high hp, so this part works definitely as expected.
It will be interesting to detect how much hps a full legendary adult mutant will have