Everyone is different, and from my observations, this question often comes from people who live in an area where the temperatures are generally quite warm and where they only experience a short period of time in the winter where it is cool, and maybe only a few days where it could be considered cold.
On the other hand, I have spent approx. 2/3 of my life living in temps that could be considered at least cool, and about 1/2 my life in temps that most people would consider cold. For me breathing in cool or cold air is normal. It's a part of every day life. I'm used to it.
The only time cold air bothers my lungs is if I do something like dash out the door and sprint for the bus ... and the temperature is about -40C with a bit of a wind. It can be a little bit of a shock to the system to go through a dramatic temperature change and to go from casually breathing in room-temperature air to gulping in freezing cold air.
However, if I decide I want to go for a ride on a day when the temps are in the "freezing cold" range, I will ease myself into it. I will open the garage door to let in the cold air while I'm getting my bicycle ready. I'll walk my bicycle down the sidewalk and across the street (I do that to avoid the piles and snow and ice too). Once I am cycling, I start easy and gradually build up my pace to something that feels reasonable for the conditions. When I do that ... I'm fine.
Now I will add that I have EIA, so damp, or cool, or cold conditions can create a certain amount of congestion, and potentially, breathing difficulties. And I'll often cough for an hour or two after I get in from a longer ride. I can avoid some of the discomfort of my EIA by doing things like starting slow, or never overexerting myself on my ride (outdoor winter cycling isn't the time to do interval training, or something like that, anyway), and especially by using my inhaler. NOTE: What I experience is not a cold, and I'm not "sick", even though I may appear to have some cold-like symptoms. It has been officially diagnosed as EIA, and a shot from my inhaler can make my symptoms pretty much vanish (or never start to begin with if I remember to use it before I go out.