I know that D♭ - E♭ is a major second. What is the interval D♭ - D♯ called (and why)?
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).
Equality is never found in consonances or intervals, and the unison is to the musician what the point is to the geometer. A point is the beginning of a line, although, it is not itself a line. But a line is not composed of points, since a point has no length, width, or depth that can be extended, or joined to another point. So a unison is only the beginning of consonance or interval; it is neither consonance nor interval, for like the point it is incapable of extension.