Is it a nickname of the Dragonborn or a weapon of the Ulfric Stormcloak?

Ulfric gave me the title of my name, but I don't know as the Hero of Skyrim or a tool of the Stormcloak.

Its just cool sounding, the nords have a thing for cool sounding titles in their name. Its just their way of giving you a cool sounding title since you're winning a war for them. The imperials give you much more official sounding titles through their questline

Thanks to my m,ilitary days I'd rather be called Stormblade over Lieutenant any day. No one ever wants to follow a Louie into battle. Someone named Stormblade on the other hand...I know what I would choose.

Actually, it's a sword: http://elderscrolls.wikia.com/wiki/Stormblade

But you could just: http://tamrielvault.com/group/character-building/forum/topics/character-build-the-stormblade

I always took that as a bit of both. Ulfric bestows upon you a cool nickname while considering you his personal weapon.

[EDIT]

Welcome to the boards!

Thank you! Nice to meet you too.

Welcome to the forums, here is your fishy stick http://www.uesp.net/w/images/c/c4/Fishystick.jpg

It's essentially just a title, I prefer legate though.