I would love a mount support champ.
The riding trait might have to be his passive(with a brief wind up like Quinn's ult), as it would feel bad to have to spend your first point on the skill(at least it would to me. Not to mention, how would it increase per level?). Also, as long as they are merging health pools, would their defenses be merged as in added together, or merged as in averaged? Or would only the highest amount apply? Personally, I am for taking a % of the rider's defenses and adding it to the mount. That would allow the mount to be used by all to greater effect, as a tank could jump on and make an unstoppleble juggernaut(though that could be an issue in and of itself), or an adc could go on for extra defense. Though the additional health and the mounts mr/ar would probably be enough for the defense boost. Maybe having the riding be a perma-passive on the ult to prevent the abuse in the laning phase? Having the mount have 340 ms, then gain base ms through the ult's passive would be another solution, as having an Ashe moving anywhere from 35-10 ms faster than the other adc from the start would be rather... unfair, especially with the range and the slows.
Life steal and regen would be another thing. If their health pools are merged, then are both healed by life steal, or would knights vow be necessary(speaking of which, since they are merged, would the passive be constantly on)? Are their health regens' merged and applied to both? Say if a health regen building Himerdinger rides the mount, does Himer's health regen fill their health, or does it only keep them at the min of 1/2 hp? If Soraka heals them, does it apply to the mount first, then to the rider's pool after it, Only the mount like kled, or does it heal both equally?
One thing that they probably would have to do is give the rider a "cannot move or be displaced" effect, to allow things like a Caitlyn net or a Graves ult to be used without throwing them off the mount. But it would call into question what happens when a spell that has a channel is used. Would Malzahar's ult or a Sion q(that's his slam, right?) immobilize the mount? Or would it allow for re-positioning of the skill? The first seems more balanced...
Ms would also be something to tackle. I think that 350(tied with Jax, Shiv, Spider form Elise and a few others)-360 would be a good base range(or the modified range with the passive on the ult), as this person is supposed to be a beast that would give extra mobility to those who ride it, and having it have the same ms or lower than a good number of champs would defeat the purpose thematically.
The only issue with the Damacia Wyverns is that it might step on Shivana's or Kled's toes too much. Her design is of a wyvern, what with the normal back legs and winged forelegs, and Skarl looks like how a wyvern is usually depicted, only derpier. Maybe those similarities aren't enough to block the design, but it bears consideration.
Side note: next champ was said to be a champ that works best in a duo lane, but is not a enchanter nor was specifically said to be a tank. So one can hope...
P.S. I relize that this might not be easy to read, but I wrote parts as they occurred to me. And my thoughts tend to be all over the place...