Compression isn't just gain adjustment - it's a specific type of audio processing that increases perceived "gain" (loudness) of the entire source audio by "compressing" the levels of loud frequencies & increasing the levels of quiet frequencies.
Normalization increases gain of all frequencies at any given point-in-time while reducing gain of all frequencies at other points in time. It doesn't reduce dynamic range.
I'm not sure what the scare quotes are about but if you point out what I'm misrepresenting I can try to explain it a little better.
Compression acts on the amplitude of the entire audio signal, not "quiet frequencies". You can start here: https://en.wikipedia.org/wiki/Dynamic_range_compression
Normalization as it normally is used is offline process where you up the gain of the whole audio so that the loudest noise is 0dB.
Normal compressors do effectively do gain adjustment - it's not really the same as a typical amp since their core function only reduces gain, then makeup is applied to the entirety to compensate - but yes the result is effectively gain adjustment.
As for doing it "without regard for the frequencies present", if you compress a mix with a base guitar & high vocals, the impact of the compression on the base will be different than on the higher notes. This is aside from (/in addition to?) attack & release applied on a per-track basis & more just about the natural effect of dynamics within frequency ranges.
Other than that I'm not sure what else you could mean. Maybe we've been using different compressors. I've used a small handful of hardware and software compressors and haven't found them to sensitive to spectral content when used "normally". Meaning no extra filters or side chain configuration.
When you do that then the difference between the loudest and the quietest part of the audio gets reduced. That's dynamic range reduction.
While normalization is usually one-way, if you're doing DSP normalization & have a record of the level offsets you've applied, it's reversible. This is never the case for compression - you can't increase dynamic range of a compressed file (short of AI-generating something that never was)