Up front cost is higher, but the efficiency gains are huge, and they can cool/heat much faster.
The only place in this wide world I’ve seen anything other than split units is North America, so “Euro style” seems an odd description.
I was using "Euro" only to help people picture the things, it's a common way I've heard them described.
Is it because HVAC is so common it usually makes sense to plug the AC in the HVAC, and the rest of the market is basically leftovers which get to have window units or crappy portable air conditioners?
In the northeast where buildings are generally the oldest in the country, window units are common - especially in rental units, because the tenant typically provides the AC themselves. One less thing for the landlord to maintain.
The northwest, for the most part, doesn’t use AC much with their climate. Many homes don’t have AC at all.
Mini-splits are becoming more and more common in big cities here now, but only if you’re lucky enough to own a place and you don’t have landmarked facade problems. I’d love to use one instead of my window units, but I rent a street-facing apartment in a landmarked building. Even if I could get my landlord to approve me paying for a mini-split to be installed, I couldn’t put anything outside. The commercial tenants in my building have mini-splits, but they have access to a side of the building which doesn’t front onto the public street, so no landmark concerns.
(The fact that many landmarked facades have a random collection of window ACs hanging off the front seems much uglier to me than a bunch of professionally-installed identical mini-splits compressors would be, but, c'est la vie...)
It's the the large horizontal vents protruding from the wall you see in pretty much every low-rise (2-3 floors) residential buildings ("apaato"), or if you go to the back of the building you can see each flat's external unit, generally wall-mounted next to or below a window.
They're almost all reversible heat pumps, though their heating capacity tends to be lower than their cooling capacity.
Note that mini splits can also use a more discrete slim duct (as the link notes), but in my experience the wall mount is by far the most common, as that's way simpler and more convenient for multi-unit buildings.