It’s not that a speaker is designed for a specific genre---all speakers should be designed and engineered to reproduce the signal they are fed. It’s that different genres of music CAN reveal weakness in a speaker that another genre is less likely to reveal. I’ll say it again: read what Ken Kantor said about how different aspects of reproduced music are effected differently by various speakers depending on the weighting of priorities chosen by the speaker’s designer. ALL loudspeakers are the culmination of compromises and design choices made by their designers.
As an extreme example, if you listen exclusively to, say, string quartet music, a speaker than can reproduce that music very well may not necessarily be able to reproduce a full symphony orchestra as well as it can the quartet. A speaker than makes a good recording of a Bluegrass group sound very much like the group does live---say the old Quad---might be unable to do the same for an AC/DC recording.
The fact is, one "kind" of music may make more obvious the weaknesses in or limitations of a speaker than may another. The ability of a speaker to keep the individual threads of notes played simultaneously by four harpsichords (as in J.S. Bach’s incredible Concerto for 4 Harpsichords and Orchestra) separate from each other is a very different task than is reproducing the lowest pedal note (16Hz) of a pipe organ, or Jimi Hendrix’s (note spelling ;-) Strat into a Marshall stack at 105dB (believe me, live it was not a "pretty" sound---kind of like chewing foil!).