You’re talking about adiabatic heating, which is where temperature changes due to change in pressure, without heat transfer. If we thermally isolate the gas as we compress it, the temperature will rise.
We don’t have to insulate it. We can allow the heat to transfer out of the gas as we compress it. Heatsinks on the pressure vessel will pass the heat from the pressurized gas into the ambient air until their temperatures equalize.
Since we can add or remove heat from the gas after it is compressed, the temperature of that gas is only relevant if it falls below the boiling or freezing curves, allowing the gas to condense into a liquid or solid.