I'm surprised this didn't come up sooner... but oh well...
Depending on how commonplace it is for wizards, spellcasters and magical artisans to create artificial homunculi, golems and constructs... and animate them (in some cases, granting them sentience similar to how intelligent items work) to function as guardians and servitors... you could then perhaps extend this concept to fashioning these creatures into forms that can be ridden as mounts. Now, take the concept a step or two further, and imagine modifying one of these creatures, such that it could bend and warp itself to surround, and "engulf" another being, form a mind link with it, and use that link to respond to that being like an extension of its own body.
Sure, I grant you its not a good fit for many of the more classic medieval fantasy settings, such as LOTR for instance.... but D&D is a strange collaborative creation that draws from many sources of inspiration.
As far as working as an extension of the body... some of these D&D settings, you have symbiots (typically aberrations bred with purpose) that can attach themselves to their host, whilst providing them some manner of additional functions or outer protections, many of which respond to the host's will. Now sure, they are not as artificial as a construct might be, but symbiots could be considered in the realm of "sci-fi".
Then you have grafts, where tissues from other beings are grown or implanted into another... or full body parts attached to another outright, (including available parts from golems, undead, plants and even elementals.... some of which involve the risk of rejection or some other malfunction as they are used).
Then you have aberrations, far realm denizens and Lovecraft-style abominations that, in and of themselves, can look pretty alien, in an almost "sci-fi" manner.... yet these horrors still find their place in some settings considered medieval.
Then you have settings that involve things like spelljamming, which involves taking open deck vessels (and sometimes whole cities) into space. In these cases, as I understand it right now... when the ships/cities part with a given planet's atmosphere, they draw off and take with them a bubble of that planet's atmosphere, kind of like blowing bubbles that pinch themselves free from a film of soapy water. And as far as space travel itself goes, one prominent example of it existing in "medieval" fantasy... Warcraft's Draenei.... who would probably be spelljammers if they were adapted for use in D&D.
However, I have to admit, that games with many developers, like Warcraft and D&D, diverge heavily from single author sources of classic fantasy literature. So, if having any of the aspects discussed above, sours the purity of that RP experience that you are looking for, then by all means, let it be known and perhaps I can find a way to bow out.