It is (slightly) hotter in one sense than still air would be. That’s why friction warms things up: it’s converting the movement of objects or fluids in motion into the random motion we think of as “temperature”.
But the molecules in air are moving much, much faster than a typical wind. The nitrogen in room-temperature air is moving at an average (well, most common) speed of a little over 400 m/s (900 mph); no wind on Earth comes anywhere remotely close to that speed.
The very strongest tornadoes might get a third of the way there, which (because temperature is proportional to speed squared) is the equivalent of increasing the ambient temperature by a factor of about 77% ((4/3)^2 = about 1.77). For everyday warm air at about 300 K (we need to use Kelvin here since we’re talking about absolute temperature), that’s enough to bring the air effectively above boiling – but air is a very poor conductor of heat and you feel air temps like that moving something out of an oven all the time without any harm.
For a more typical stiff breeze of, say, 10 m/s, it’s a difference of about 5% (that is, (410/400)^(2) – 1) in terms of absolute temperature, or about 15 K in typical 300 K warm air. You might notice that if it were in isolation, but the wind blowing over your skin will cool it much more than the extra motion of the air will warm it up, because it’s blowing away the thin boundary layer already warmed by your body heat and moistened by evaporation.
Temperature is “the atom moving” in the sense of how fast the electrons swirling around the middle, not the whole thing moving.
You could have a stone-cold pile of atoms screaming through space at near light speed.
You could have a super-hot atom at a stand-still at the core of a reactor.
For gases specifically, when the electrions swirl about faster and they spread out, pushing other atoms away. That makes the whole thing less dense, lighter, and that’s going to cause it to rise and move about and stuff.
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