Is there any specific reason why Delta Air Lines doesn't have its own callsign?
Delta Air Lines uses the ICAO three-letter designator DAL and the ICAO telephony designator (also known as callsign) DELTA.

In general, callsigns should be similar or equal to the name of the airline according to the following ICAO rules:


  3.2 In the registration of telephony designators the following rules will apply:
  
  a) the chosen telephony designator should attempt to resemble the name of the aircraft operating agency or its function
  and be distinct and dissimilar from any other telephony designators in Doc 8585. Ideally it should reflect correlation
  between the three-letter designator, the telephony designator and the name of the aircraft operating agency or its
  function (examples: ARO – ARROW – Arrow Aviation; RAJ – RAJI – Raji Airlines);
  
  b) in order to reduce the length of transmission the telephony designator should be brief, comprising if possible one
  word of two or three syllables. It should not exceed two words;
  
  c) three-letter designators may not be used in phonetic form as telephony designators. However, telephony designators
  of long standing (such as KLM or TWA) may be retained, provided that an acceptable alphabetic representation is
  used (example: KAY-ELL-EMM); and
  
  d) the telephony designator should be easily and phonetically pronounceable in at least one of the following languages:
  English, French, Russian, Spanish.


(ICAO Doc 8585 3. Telephony designators, emphasis mine)

Different callsigns can exist for two reasons:


Old callsigns: Several old airlines use callsigns that existed before these rules were made, like e.g. KLM as mentioned in c). British Airways inherited the callsign SPEEDBIRD from BOAC in 1974, which in turn inherited it from Imperial Airways in 1939. It is named after the design of the Speedbird logo.
Conflicting callsigns: When the airline name is too similar (or equal) to an existing callsign, a different one must be chosen, like e.g. Norwegian mentioned in J. Hougaard's answer. US Airways inherited CACTUS from America West in 2008:


  Early in its history, the airline used the call sign “America West” but it often caused confusion with other airlines ending in “west” (Southwest, Northwest, Skywest). The FAA suggested that America West come up with a new call sign. The company held an employee contest to come up with the new name. Cactus was chosen.


(aerosavvy.com)


While Delta Air Lines is quite old ("Passenger operations began on June 17, 1929", Wikipedia), it did not chose a different callsign. I can only speculate on why, but note that the ICAO phonetic alphabet is only officially used for civilian aviation since 1 April 1952. Before than, other alphabets were common, including the Allied military phonetic spelling alphabets, which used DOG instead of DELTA, so there was no conflict at the time.

Using a different word when confusion is likely (like DIXIE in Atlanta), is explicitly allowed in the US:


  ATC facilities may also request pilots to use
  phonetic letter equivalents when aircraft with similar
  sounding identifications are receiving communications on the same frequency.


(AIM 4−2−7. Phonetic Alphabet)