Harnessed Quintessence:
You master the warp and weft of your very soul. When you are restored Magicka or Stamina, increase your Weapon and Spell Damage by 5% for 10 seconds.

This does not normally proc from recovery stats, a tick of stamina recovery does not proc this passive. However, it appears that if you have less than full stamina mounting your horse will cause it to proc. I suspect this is because the "full" horse stamina is being counted as a "stamina restore" when computed against the not-full player stamina.
This does not occur when mounting when the player's stamina is full, only when the player's stamina is partially depleted (and does not tick to full during the mounting animation).

Mounting with partial magicka does not proc the passive. Dismounting with depleted mount stamina does not proc the passive. Only mounting with depleted stamina.

Not a serious bug, just a really weird one.

Edit: As discussed below, this is an interaction between the Altmer's Spell Recharge passive, and the Arcanist's Harnessed Quintessence passive. Summoning a mount is an ability, casting that ability can proc Spell Recharge if you have depleted resources, if resources are restored from Spell Recharge, then Harnessed Quintessence will proc.
I just didn't see it with partial Mag before, because I didn't actually drain it to below my max Stam value.

This can be closed.