So the fourier-transform of white noise is still.... white noise. Random is random as you say. But this has implications. That means the "wattage" of noise (ie: Voltage * Current == Watts aka its power) is a somewhat predictable value. If you have 0.5 Watts of noise, it will be 0.5 Watts of noise in the frequency-domain (after a fourier transform, across all frequencies).
The hard part of amplification is keeping it consistent across all specifications. I assume the previous post was talking about keeping white noise (which is "flat" across all frequency domains), truly flat. IE: It means your OpAmps (or whatever other amplifer you use) CANNOT distort the value.
Which is still student level (you cannot be a good EE / Analog engineer if you're carelessly introducing distortions). Any distortion of white-noise is easily seen because your noise profile weakens over frequency (or strengthens over frequency), rather than being consistent.