This isn't universally true. Different tunings have been used in different time periods and locations. The modern standard pitch A440 (A4 at 440Hz) was only defined in 1926. Before that, the "standard" pitch has been as low as 415Hz (baroque pitch) or as high as 465Hz (Chorton pitch).
From https://en.wikipedia.org/wiki/Concert_pitch :
> For example, a 1740 tuning fork associated with Handel is pitched at A = 422.5 Hz, while a specimen from 1780 is pitched at A = 409 Hz, about a quarter-tone lower. A tuning fork that belonged to Ludwig van Beethoven around 1800, now in the British Library, is pitched at A = 455.4 Hz, well over a half-tone higher.
Besides the chosen pitch, only in an equal-temperament tuning can a piece of music be transposed to a different key and sound identical. Under different tunings, the ratio between subsequent notes is not always constant so transposing a piece of music to a different key will produce slightly different harmonic intervals.