@Missezem Try this: As with many Jackspeaks the origin is shrouded in the mists of time but my understanding is that the original Donkey Engine was a small, winding, steam engine ashore which replaced the Equidae in various applications eg logging. In the merchant service a small, steam, auxiliary deck engine (Crane/winch etc) was/is often referred to as a Donkey Engine whilst the RN applied that term to auxiliary diesel engines, as opposed to the steam and later gas-turbine main engines. I believe the term Donk Shop thus evolved into RN useage as the term for the diesel engine room.
Re: the stoker as a 'horse' I'd suggest merely a play on words due the passing similarities of the donkey & horse. (Heard elsewhere in the RN where a Telegraphist was sometimes described as a 'morse-horse' purely for the rhyming value.)
Gen Dit: I once served with an Marine Engineering Artificer with the name of D. O'Brien. As his role ATT was the Donk Shop Horse his nickname of 'Dobbin' needs no explanation (his 'rigging' ticked the right box, too. )
To while away the middle watch on Mermaid myself and Figgy Duff the DG/VCD w/keeper would stand on the coupling casings of the 2 V16 ASR 1 main engines and re-enact the scene in Ben Hur chariot race , me Stephen Boyd and him Chuck Heston imagining the 'horse' power in front of us to be real horses......stokers eh