Q: Does all "white noise" sound like the same hiss?

The answer is no: not all white noise sounds alike!

White noise can sound like "hissing" of a shortwave radio or it can sound like a Geiger counter (click..... clickclick............ click).

More:

A noise process is "white" if every frequency has the same power spectral density.

Any process where any two samples taken at different times will be statistically independent is white in this sense. In other words, if knowing the amplitude of the noise at time x tells us nothing about the amplitude at any other time, then the noise must be "white."

But there are many different-sounding processes that have this characteristic, because just knowing that two samples are independent does not tell us the distribution of the individual samples.

• One classical example is "thermal" noise, in which the samples are distributed according to a normal, or Gaussian, distribution. This is known as "white gaussian noise," and typically in communications will have been added to the signal we are interested in: hence, Additive White Gaussian Noise (AWGN). This sounds like "hissing."
• Another kind of white noise is "shot" noise, which can come from any Poisson process, including the particle decays heard by a Geiger counter. Here the individual samples aren't Gaussian deviates; they are impulses, either zero or big, and most of the time they're zero. But since knowing the time of one "click" tells us nothing about any other (and because each click carries all the frequencies), this is also white noise.