Yes, an airing cupboard is a feature of older houses, but since most - or at least a lot - of us are still living in older houses, they’re a common feature.
(That’s a surprisingly difficult figure to look up, but basically it’s just not uncommon at all to live in a house built at least 50 years ago).
It’s literally a cupboard. It has no windows and isn’t large enough to walk into.
I turned mine into a “butler’s pantry,” with a washing machine/tumble dryer combi, a mini fridge on top of it, a dryer rack on the door, and shelves above it all (it’s much smaller than that makes it sound). I’m not entirely sure why clothes in there dry so well - assume it’s to do with the washing machine venting air - but they do.
Otherwise I have a rack fitted to the hallway wall, above a radiator, plus racks on top of radiators, for times when drying outside isn’t possible.
Combi washing machine/tumble dryers aren’t really supposed to be able to dry the same load they can wash - there just isn’t the space for air to circulate within the dryer, and it takes forever, and costs a lot to run. So I always take out jeans, jumpers/hoodies, blankets, towels, and bras, then tumble dry what’s left. But 90% of the time actually I take it all out and hang it on those various drying places. The dryer is still handy to keep up with the laundry when it builds up.
In the summer I don’t use the tumble dryer much at all.
I think the UK and US are pretty different in their habits when it comes to laundry, though. I know apartments in the US are more likely to have their own washing machines these days, but in the UK that’s a sign of extreme poverty - there will always be fittings for a washing machine in every apartment, so if someone doesn’t have one, it’s because they can’t afford the actual machine. OTOH, they’re much smaller, they’re front-loaders, and it’s less common to have a separate dryer, even in a house.