That statement is not true of Western music in general, only of most (not all) classical and poular music since roughly the 18th century, when "equal temperament," tunings designed to make all keys sound the same, became popular.[1]

Much Western music, such as that written during the the Baroque, Renaissance, and medieval eras, as well as certain modern genres like barbershop quartets, modern classical, electronica, microtonal and atonal music, etc. use a variety of other systems such as some form of just intonation[2], where different keys sound very different as they exhibit differently sized versions of the same interval.

Conversely, much non-Western music also uses some kind of equal temperament (where keys all sound the same), such as Chinese, Arabic, Indonesian, Thai, and Native American music.[1]

[1] https://en.wikipedia.org/wiki/Equal_temperament [2] https://en.wikipedia.org/wiki/Just_intonation