I know that Db - Eb is a major second/diminished third. What is the interval Db - D# called (and why)?
EDIT: oops, Db - Eb is NOT a diminished third.
D-D is unison (or "prime")
Db-D or D-D# is augmented prime and Db-D# is a double augmented unison or prime.
P.S. This is really something different than the enharmonic variants Db-Eb, but that was not the question.
Consider also the questioning of naming the unison an interval by italian music theorist Zarlino which I personally find very relevant (rather think of it as a point in geometry).