It's not compressing air (like in a car tire). It's compressing a refrigerant. That refrigerant goes through phase changes (liquid to gas).
One major issue is that for most ACs, the compressor is cycled on and off according to the target temperature (via a thermostat, usually at a single location), not humidity. That means humidity can rise without the AC kicking on to bring it down. Remember in most typical houses, temperature and humidity are not very uniformly distributed.
Furthermore, if the humidity rises high enough before the AC kicks on, and then the AC kicks on at high power, you can get sudden localized cooling and then condensation of humidity to liquid water inside the building, which leads to other problems, especially if it happens behind the walls.
It's possible to independently add humidity when heating -- using a mist gun -- but not to remove it during cooling. However, if the heat pump has a "dry mode" it can dehumidify without also cooling by switching back and forth between heat and cool mode. If not, to dry the air further, it must cool it further.
So when the unit runs it can drop the temperature by the necessary (e.g.) 5F (2C), but it may only drop the humidity by 5%, when it needs to drop by (say) 10%. So a 'too-short' run-time can adequately cool the air, but not necessarily remove moisture.
It's also easier to generate 'excess' humidity by bathing/shower than it is to generate excess heat (cooking could generate both). So the humidity can creep up in value while the temperature stays more steady.
But yes, I have a heat pump and in NYC Summer I cannot run it on anything but low otherwise it increases the humidity. It took me a few weeks of looking at the temp humidity graphs to understand that point.