Music can have a mental effect on animals as well, exemplified by the study of Uetake et al. (1997) who demonstrated that dairy cows exposed to soothing music are more willing to be milked. There are various accounts of animals, such as birds, dogs and apes, that take a particular interest in music, but dogs who "join in" when people sing probably do so because they have a tendency to howl with the pack rather than because they enjoy the particular song. Likewise, those species of birds that have an innate tendency to imitate sounds may imitate music, but this is probably not driven by the kind of enjoyment we feel.
The more common effect seen on animals is one of music being soothing or relaxing. It seems fair to assume that the intense emotional response to music is a specific human trait, which probably evolved during the last few million years.