[...] but the animals in question would likely need to be properly trained for it as well. At this point there might as well be a new building for it, the Stable.
My first thought when I read the suggestion was a Tiltyard. Not as a Building, but instead an item-defined Room. If intended to be a solo training effort, a quintain could define the area (equivalent to an archery target, inclusive of direction-of-approach style specification), or if you want a sparring-equivalent partnered practice then a row of hurdles laid down or specified similar to horizontal axels. Why not both, though? Or a "riding the rings"/"tent-pegging" linear feature for one? Or sequential archery targets used for high-speed mounted-archery practice so your bear-mounted crossbowdwarves can work on the rather difficult to advance mountedmarksdwarf skill?
But Stables as a building either: a) to more realistically house non-wandering mounts (equine, ursine or whatever) than 'infinicages'[1]; or b) to take over "train a war/hunting mount" from the Kennels... now, that's a further idea.
[Insert: I'd suggest a Livery Stables speciality building/roomplan for chariot/beasts-of-carting training/housing/whatever-one-ends-up-doing-with-it. It would mean things weren't lumped together
too closely in use so that if either type of stable gets implemented on its own merit, it needn't require/preclude the other type.]
Though as there is Creator-resistance to this idea, I'm speaking purely speculatively, here. [Inclusive of that last insert!]
[1] Though a Barracks-style room-definition might again be more suitable. One stall (either pre-built furniture item, a-la-bed, or log-sourced on-the-spot creation like the windmills) needed per currently non-wandering mount to avoid badthought-style 'problems' with our four footed/hooven/flippered friends. And additionally the need to be baled-up with/within range of store of longlanded grasses/equivalent prefered food matter and drink (H2O, CH2OH, SiO2(liq)[2] or whatever else takes their fancy) might be 'interesting' (FCVO...) and provide a balance of difficulty when maintaining ones steads.
[2] We're going more and more into the fantastical, here, but imagine the choice of "(b)uild, (z)table" [sic, tr/z/s/] or "(b)uild, Magma (Z)table" [likewise]. And the mounts that require the latter.