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I don't really know the Dutch tradition; I never encountered it like this in Germany, but I suspect that it probably exists close to the border with the Netherlands. I only knew the card game "Schwarzer Peter":
https://en.wikipedia.org/wiki/Black_Peter_(card_game)
It's interesting that there does seem to have been racist versions of it, although the rules must have originated much earlier, and as people developed more racist attitudes, this game was an obvious target to attach them to it. When I was young, the only version of a special set of cards for it that I saw had the Schwarzer Peter card as a chimney-sweep:
Black Peter has long been a popular children's game and numerous proprietary packs are produced today aimed at the children's market. In older packs, the Black Peter was typically a stylised image of a Black person; more modern packs use a variety of images such as chimney sweeps, black crows or black cats.
I suspect someone should edit that Wikipedia entry, too. I've changed the above quote slightly.
"Den Schwarzen Peter haben" ('to have/hold the Black Peter') is proverbial in German for 'to be stuck with a negative' (as you lose the game if you're the one left holding the Black Peter card at the end), as is "jemandem den Schwarzen Peter zuschieben", which means 'to pass the Black Peter or to foist it on someone", i.e. pass a problem on to them so that they have to deal with it instead of the person passing it on.
https://de.wikipedia.org/wiki/Schwarzer_Peter
Famously, Germany had few colonies, as it only acquired them after Bismarck was out of the picture (although that wasn't the beginning of German, or indeed European, influence in southern Africa, see the article below), who strongly opposed the 'acquisition' of colonies, but in the short time it had them, still managed to commit genocide:
https://en.wikipedia.org/wiki/Herero_and_Namaqua_genocide
The racist element may have been stronger in the Netherlands (and perhaps Belgium) with their stronger history of colonialism, and the way you experienced it was most likely still coloured by that.
The notion of a helper of Father Christmas ("der Weihnachtsmann") who goes down chimneys and gets soot on his face doesn't exist in Germany, as far as I know (although invariably it'll probably exist somewhere). In Germany, it's generally Father Christmas himself who supposedly goes down chimneys (or not infrequently gets stuck). In Germany, the assistant to Father Christmas is said to be Knecht Ruprecht, a fairly bizarre character who doesn't have any connection with racism, though:
https://en.wikipedia.org/wiki/Knecht_Ruprecht
Needless to say, there's still plenty of racism in Germany, just in other areas.
I still think black Pete is the nastiest example of this I have experienced. My second trip to the Netherlands I had a friend of a friend try to convince me that it's all just part of tradition and it hasn't harmed anyone so it should be kept because the kids love it. I know it has been toned down recently but I'm pretty sure black Pete still features in a lot of Xmas events over there.