Ulysses wields a non-playable version of Old Glory. The weapon has higher damage, but lower critical damage and a critical multiplier of 1. It also has a 10% chance of knocking enemies down (unless they have Stonewall). The weapon scales to the player character's level, gaining extra damage and critical damage at levels 25, 35, and 45.
Old Glory was originally meant to be a weapon in Fallout: New Vegas, then named the "Eagle Pole". It was cut from the base version of the game as a usable item, but its visual resources remained intact.
The 3 of Clubs in the Fallout: New Vegas collector's edition card deck depicts Ulysses holding what appears to be a flag pole.
It is possible to get two copies of the weapon; if you finish The End with Ulysses still alive he will be sitting on a rock ledge, over looking the The Divide after you return back from the Mojave Wasteland. Alternatively you can just kill Ulysses before completing The End/The Apocalypse, take it from Ulysses' corpse and another one will be in the footlocker waiting for you.
The name "Old Glory" is a nickname of the flag of the United States.
The idea of carrying a gold or silver pole with an eagle on top dates back to the Legions of the Roman Republic in which each legion had two "Eagles" which were to never be captured. They symbolized the courage of the members of the legion. Since Ulysses was/is a member of the Legion, he may have been an eagle bearer, showing Caesar's insistence on imitating Rome.
Ulysses' dialogue in Lonesome Road indicates his eagle staff is an Old World symbol.