as here they imply simply bringing leftovers home, although it has been a very long time since I've heard the term doggy bag and am more accustomed to people asking if anyone wants a box or their leftovers wrapped up.
It took a while for me to get used to the US phenomenon of asking for leftovers to be "boxed up" and I've noticed a similar level of surprise with less travelled Europeans I've taken to the US.
I'm used to it now (though I still wouldn't ever do it myself) but this is not even slightly common or acceptable practice in the UK. Such a request wouldn't be met with disgust or offense, though, merely confusion. I doubt many places except Chinese restuarants/takeouts would have anything to put the food in anyway.
So, yeah, this is a very American practice from the UK POV. I'd be surprised if it weren't common somewhere in Europe, though, but I've never seen it happen on my travels.
Splitting a meal or asking for more utensils to share a dish isn't exactly common here either but it is done, especially in pubs or similarly low-end places.