AC air feels unnatural
A typical (I am sure there are exceptions for various extreme situations - perhaps clean rooms, hospital operating rooms, extreme environmental situations, space station (ISS), etc. - though it may turn out that nearly all (except ISS) are really just "regular air conditioning with higher power and/or more filtering") does the following:


Use some method of cooling to indirectly cool the air in a building. In a traditional home air conditioner that is via freon which goes through a compression/expansion cycle so that it gets very cold to go inside and cool your air and then releases the heat outside. The mechanism is a little different with a chiller, but the basic result is the same: some method gets something really cold and the building air transfers heat to that thing (e.g., freon or water) and the air ends up colder. But the air does not "touch" the actual freon or other cooling fluid.
Filter the air to remove dirt/dust/pollen/etc. That can range from a simple replaceable filter to a HEPA "super" filter that can get the really small stuff out. Some systems use an electrostatic filter to get the dust out electrically - but the goal (and result, hopefully) is always the same - make the air cleaner. Air circulated through a house repeatedly (and especially with modern buildings that are well sealed) collects a lot of dirt from people, pets, etc. so filtering is important.
Blow the air through the building. This is required for air conditioning because, unlike heating where the heat can radiate through the house naturally, cold air will not do much unless it gets pushed to where you need it.


There is a lot more variety on heating systems, but for HVAC that's all there is to it. So why does it feel unnatural?


Dry air. A natural consequence of air conditioning is that it will lower the humidity of the air. In many areas, hot summer air is also humid air ("It's not the heat, it's the humidity!") and that is because hot air can hold more water vapor than cold air. When the air is cooled, it goes below the dew point and water vapor condenses and needs to be removed (preferably by gravity, but if that is not sufficient then there will be a condensate pump). As the air warms up again, it will end up much less humid and feel different. This is good for a number of reasons, provided it doesn't get too dry. But it will definitely feel different. As an example, if you have a hot, humid day but at night if the temperature outside drops to 72 F, the inside air at 72 F will feel different because it has been dried over the course of the day when it was cooled.
Blowing air. A breeze feels different than stagnant air. While the average air speed inside a house is very low, next to each air vent it will be fairly high.
Cold air. You can't easily cool a house to 72 F by blowing 70 F air. So you blow air at 60 F (or colder). That is much colder than you are normally comfortable with, but it mixes with the rest of the air quickly and averages out to the set temperature. Similarly, in the winter (if you have forced air heating), the air coming out of the vents will be a lot warmer than 72 F but it will all average out. Some people like that feeling "I'm hot - ahh, nice cold air" and "I'm cold, ahh, nice hot air" but some people are uncomfortable with that differential.