Hiro's half correct.
The answer is: it depends.
More often than not, it's at the platoon level...but certainly not always (Company K "Wildcats" or "The Blackguards" come readily to mind).
If you check out the Roghnecks Army book you will see examples of both "official" and "un-official" logos. My thinking was that the SICON heraldry folks would release SICON-sanctioned insignia for Divisions and Regiments. Lower-echelon units were not included. Still, soldiers being soldiers, they would almost certainly come up with with unique names and emblems for "their" units...and those would tend to perpetuate themselves over time.
My Pathfinder platoon (Palmer's Parrotts) is an example. The first platoon leader (2LT M'Basa) had a pet parrot, and the platoon started referring to themselves as "the Parrots."
Historical footnote: that first parrot, "Ulysses" is widely held to have had a particular fondness for the snack food "Krunchy Kurls" soaked in vodka. Also, the unit's legends maintain that he was the only avian in SICON history to participate in multiple combat drops. The current mascot, Ulysses III, is less adventurous. He prefers to sit out drops in the comfort of the wardroom aboard the TFCT General Ney. Unfortunately for the current platoon leader (1LT Palmer) Ulysses III dislikes him intensely. Unit tradition dictates that the Platoon Leader is responsible for the care and feeding of the mascot, so it is not at all unusual to see 1LT Palmer with bite marks on his hands and face...or to hear profanity emanating from the wardroom at Ulysses III's feeding time.