Warm in its modern conventional audio usage definitely relates to frequency (IMO).
Play a pure sine wav at 65hz (C2) and 4186hz (C8) and ask yourself which sounds warmer?
They are the same wave but different sounds, so pitch is unsurprisingly a component of a given sound.
DonovanDwyer said it above, warm basically means more low frequency and attenuated high frequencies.
“Tape warmth” can be an artifact of the physical media of tape losing its ability to represent higher frequencies as it gets more worn. This would bring a relative emphasis to the lower frequencies in the material (also compression but I will ignore that here for simplicity).
It’s a bit like “dark”, which generally means attenuating the higher frequencies, vs bright, which I would say is typified by more prominent higher frequency content.
I think warmth in a mixing content basically means darker highs, presence in the low end but still with good mid range.
I also think about “color”, where its not like a literal hue, its more like describing differences in representing the same material.
For example, a deadpan FBI agent explaining that birds aren’t real vs the rough looking guy shouting about it to nobody in particularly when you get off the bus. One definitely has more color than the other. Similarly, equipment and processes can have a lot of color or be thought of as relatively clean.
The Nord Lead is a synth I owned which I feel completely lacked any warmth and basically was impossible to impart warmth with post processing. It was cold, hard, cutting and clinical. It sounded wicked with distortion as well but you were never going to get a a lush, warm pad out of it.
Warmth, color, etc arent technical terms with rigorous quantifiable definitions but adopted language that we use to describe things to other people working in the same field, who have similar, learned perspectives.