Why don’t we see icy (blueish) and hot (reddish) air in different colours?

Our eye perceives only a (very small) part of electromagnetic waves and we call this part light. This wave is described by its frequency (in visible light, the frequency corresponds to a certain colour). The colour of the body depends on its temperature (if we are talking about light sources). For example, the Sun is yellow because the temperature of its surface is roughly 6,000 °C, if we cooled it, it would be red and conversely, if we heated it more, it would be blue and then purple.

In order to be able to describe how the colour of a certain body depends on the temperature, physicists introduced a simplified model — an absolutely black body and they were able to determine a mathematical relationship for it, which determines what colour the absolutely black body (ABB) will have at a given temperature (the Wien displacement law). In order for us to see the body glow, it must have a temperature of at least 900 °C, it would have a blue colour at a temperature of 3,000 °C. I would just point out that the temperatures listed are for a simplified model of the ABB, for real bodies, the colour of light at a certain temperature depends on their composition.

I think it's a good thing that we don't perceive colour differences between warm and cold air — because otherwise we'd have to live in an environment with temperatures of 500—3,000 °C.