I confirm that it's common in bakeries and many takeout places in Germany and many western european countries.

I've seen them everywhere I've visited in Germany. An image search came up with this¹, which says they're called "Zahlteller" (counting plates).

1: http://www.schnitzelbahn.com/homepage/2010/11/25/the-evil-za... (scroll down a bit)

This is a common practice that started out in Western culture. It was, and still is, considered more polite to receive your bill and put your change on a tray. Go to an extremely expensive restaurant and you'll see. Here in America, we now prefer the billfold instead of the tray for privacy reasons, but the concept is exactly the same.

Isn't it common in Restaurants to leave the money in a tray?

Something much more weird I think is the money envelopes. I once took classes by a Japanese lady who, the day before we would pay for the next month, gave us these envelopes the size of a bill and asked us to bring the money in them, said that Japanese never hand money directly, always in envelopes. The next day she took the envelopes and didn't even open them to count the money.

You probably just didn't notice, because it is totally normal to not use them, but they they are common. I think they see the most use at places like bakeries, where high counters make handing money over directly harder.

In Finland we give the cash/receipts straight to the person. Nothing rude about it. I don't remember there ever being any hand contact involved in it. I guess some restaurants might use trays, but not the ones that I ever go to.