Whether or not any given speaker "favours a certain kind of music", there are certainly examples of speakers that are poor choices for certain kinds. A mini monitor (or the QUAD ESL) for reproducing pipe organ recordings (a Symphony Orchestra performing, say, Berlioz Requiem, such as Colin Davis conducting The LSO. Or Saint-Saens Symphony No. 3 by George Pretre and The Paris Conservatoire Orchestra with Maurice Durufle on organ. Or AC/DC’s Back In Black ;-) played at live listening SPL. Another reason to buy a DBA!
One who listens to predominantly acoustic Folk (or Bluegrass) music may not need the same "kind" of speaker as does one who listens to Death Metal, just as a driver who travels on only highways doesn’t need a car with the same abilities as does the driver who spends time on winding mountain roads. It’s the ol’ sports car vs. tractor scenario. Horses for courses: It’s not only what a speaker can do, but what it can’t. Different musics certainly DO place different demands on loudspeakers, which reward the musics accourding to their strengths and weaknesses (or abilities and inabilities). IMO ;-) .
Yes, a perfect loudspeaker would be able to reproduce all musics equally well. Since there aren’t any, and since different speakers have different imperfections (or inabilities), does it not make sense to choose one whose abilities and inabilities align well the demands the music the listener is most concerned with places on the speaker? You can’t have it all, though the "best" speakers may come closer than more compromised ones.
What abilities are most important to you? What are you willing to give up in order to get something else? I can’t live with a speaker exhibiting ANY vowel colorations (the great term coined, as far as I know, by J. Gordon Holt), And I need a speaker which can reproduce the contrasting timbre of the harpsichord, cello, violin, and flute playing J.S. Bach, revealing the very complex (and changing) root/harmonic overtone structure each instrument produces (speakers vary DRASTICALLY in their abilities to do that). But pinpoint imaging and a great soundstage are not that important to me. To each his own!