Since there were Dragonborns before the Skyrim one doesn't this mean that there is also one during Elder Scrolls Online timeline? At some point they will obviously unlock the remaining Skyrim area that is not available in ESO so maybe we will meet him/her there since its the most iconic place for such a character.
Varen Aquilarios trying to artificially make himself "dragonborn" is what messed up the world in ESO in the first place... just because he wanted to be a "proper" emperor, and as such you need to be dragonborn to light the dragonfires with the amulet of kings.
And if they had a dragonborn somewhere, they would gladly pop him or her on the throne, end the whole oblivion crisis with a relighting of the dragonfired that way... which won't happen until Tiber Septim as we know from the lore.
Although history does not record a dragonborn at the time of ESO, chances are that there is one out there somewhere and is either unaware of the fact, or hiding the fact.