The "Unit" nickname is not unique to Delta.
While my son was in the 160th SOAR his 1st Battalion helos (aka "The Show") routinely transported Deltas, SEALS, CIA, DEA and similar critters - collectively, the 160th's "customers." My son emphatically corrects me if I ever call Delta a special forces unit. No, he says, "Special Forces" refers exclusively to the Army's SFOD-A, aka Green Berets.
Many use the terms "Special Forces" and "Special Operations" interchangeably. Those terms are not interchangeable. Deltas are Special Operations assets, while Green Berets are "Special Forces" assets.
Deltas are officially "1st Special Forces Operational Detachment Delta" (SFOD-D), and are informally nicknamed Delta, CAG, etc. Special Operations (Spec Ops) is an umbrella term used to describe all of the US military's "Tier 1" (elite) special-mission groups: Deltas, SEALS, Air Force Pararescue and Combat Controllers, etc. SFOD-A is not a Tier 1 organization, but rather Tier 3. Still vital, but a very different mission profile.
Some believe the nickname "the Unit" exclusively describes Delta. It does not. The 160th SOAR, for example, is also referred to as "the Unit." Actually, the term "Unit" responds to the lack of traditional command hierarchy. The 160th SOAR, for example, is a Regiment - an organizational structure that does not exist in any of the Army's Division commands (e.g. 1st Cav, 1st Armored, 1st Infantry, 101st Airborne, 82nd Airborne, etc.). The 160th does not report to a traditional Army Division, but rather to JSOC for operational purposes and to the US Army Special Operations Command (USASOC) for administrative purposes (pay and promotion, etc.).
JSOC operationally (command and control) "owns" all of the service branches' special operations assets - Delta and SEAL Team 6 (counterterrorism), the 160th's 1st Bat assets (air mobility for all Spec Ops missions), AF Combat Controllers (Spec Ops mission coordination with USAF assets), etc. JSOC has the priority right to use all other DoD assets to support a JSOC special mission. So Army Rangers, MARSOC Marines, all other SEAL teams, SFOD-A Green Berets, all Army Combat Aviation Battalions' helos, etc. can be deployed to support a JSOC special activity. [Example: during the bin Laden Pakistan mission, Army Rangers stood by in Afghanistan in a 160th Chinook, ready to mobilize should the raid have required ground support.]