I think you're confusing things a little: we're not talking about restaurants per se here.
I don't know whereabouts in Europe you are, but you have to imagine a store like H&M where you generally place goods on a countertop to purchase them. In Japan, those countertops usually have a nondescript tray also lying on them, and it is considered impolite to hand your payment directly to the cashier--instead, while putting down all of your goods for purchase, you simultaneously put your payment into the tray.
There are of course restaurants where they'll hand you a bill and you bring it up to the counter -- at least in the US. For example, in the Kansas City, Missouri region you'll see this both in the sit-down tex-mex chain-restaurant El Maguey as well as smaller sit-down cafes (the one that's coming to mind is City Diner, which is by the River Market). But that's not quite what we're talking about here.
I don't know whereabouts in Europe you are, but you have to imagine a store like H&M where you generally place goods on a countertop to purchase them. In Japan, those countertops usually have a nondescript tray also lying on them, and it is considered impolite to hand your payment directly to the cashier--instead, while putting down all of your goods for purchase, you simultaneously put your payment into the tray.
There are of course restaurants where they'll hand you a bill and you bring it up to the counter -- at least in the US. For example, in the Kansas City, Missouri region you'll see this both in the sit-down tex-mex chain-restaurant El Maguey as well as smaller sit-down cafes (the one that's coming to mind is City Diner, which is by the River Market). But that's not quite what we're talking about here.