Music included on a dvd-video will be part of the main audio stream, and generally not selectable separately from the rest of the sounds in the movie. There are two basic ways of getting the music removed from the other sound and both have been mentioned already.
You can select a specific section of the movie where the music is playing and save it as a separate dvd-video. Then you demux and save just the audio. DVDShrink in ReAuthor mode will let you cut out a specific section. You can then demux that small dvd-video with any number of different tools, like Rejig, TMPGenc, PGCDemux, etc.
The other way is to remove the audio stream in its entirety using DVDDecrypter in IFO mode and then edit the audio stream, cutting out the sections that you want to save. The audio will probably be ac3 format, though it could be lpcm, mp2 (PAL) or DTS.