I thought he was Dragonborn at first with the way the Greybeards were dancing around whether or not I was the only one. It was actually a factor in me joining the Imperials for the Civil War; I wanted the fight to be fair.
Gotta second this; the first couple of quests in the chain, at least, are eye-gougingly stupid. And pretty much all they had to do to improve it by leaps and bounds was just not require the player to become a vampire slayer right before sending them to a vampire they aren't supposed to slay.
I think he might have thought he was Dragonborn at one point, before the Dragons came back and made disproving it trivial. You'll note Delphine and Ulfric recognize each other, and also don't seem to be terribly happy with one another. Also Delphine knows what the Greybeards' check-for-Dragonborn test is, despite the last verified Dragonborn dying some two hundred years prior.
So I think what happened is:
- Ulfric started training with the Greybeards.
- Ulfric ended up being, for a non-Dragonborn, very good with the Thu'um.
- The Greybeards, not having an existing Dragonborn to compare with, decide he might be Dragonborn and send him after the Horn of Jurgen Windcaller to test him.
- At some point during the above, Delphine and Ulfric meet, possibly she helps him out a little bit, thinking he might be Dragonborn.
- Ulfric fails the test, they part ways.
This also explains why Delphine wants to see you eat a Dragon soul before she'll believe you're Dragonborn, rather than take the Greybeards' word for it.