Yes, at higher speeds (like towards the end of that recording once the link is fully established) dial-up modems sound like white noise.

There's an interesting reason for that. A digital modulation that is going fast enough, if it is well-designed to make use of a channel, will approximate white noise. A primary frequency or harmonic can be thought of like a "hot spot" and the more relatively powerful it is, the more it will distort the other frequencies. So, an ideal encoding will have constant average power across the entire bandwidth. Essentially, white noise.

Modems got there by the late 1980s with the jump to 9600 bps, when a single bit started being spread out over many symbols using convolutional modulation.