Delta DOES have its own callsign. It's "DELTA".
There are literally thousands of airlines in the world, many of which have designated radio telephony callsigns. Some of them are very close or even identical to the airline name, others are more diverse.
Bianfable gives a good explanation regarding the origin of the Speedbird callsign. While not the case with Speedbird, there are examples of airlines who needed to choose callsigns very different from their name simply because the other callsign was already in use by someone else. One example is Norwegian. They use the callsign "Nor Shuttle" (and a few others), probably because the callsign "Norwegian" was already in use - by the Norwegian Air Force. But, generally speaking, a callsign will be close or identical to the airline name.
A full list of airline callsigns is available in ICAO DOC 8585.