I had to look up cashier's check and it does not sound familiar at all. But searching a bit further it is a thing that exists, seems to be called Bank Check around here, it's just kind of expensive to use.

It's much more usual to pay in cash or use an account transfer (SEPA transfer) which is usually free, but with the delay of the transfer one of the parties usually takes a risk.

This year, in relation to the rule the post is about, they also forced banks that were dragging their feet to start supporting instant transfers.

I don't ultimately know the answer to your question though. I suspect the banks have to ask you for the origin, as if you turned up with 10k in cash, but I couldn't find anything definitive in the time I was searching around.

I can't speak for all European countries, but at least in the Netherlands they're not a thing. We might still have money orders, which are similar, but I've never seen one used.

Instead we can just make instant bank transfers, even using a QR code, which you can generate in your own banking app and can be used with any other bank.

Cashier’s checks existed in Belgium a few years ago but I heard they are under fire and will be discontinued at some point.

Personal checks seem to be non-existent but I heard they can be requested but the banks give some resistance and try to steer people away from it. They only work domestically. I think if you gave a Belgian personal check to a Belgian, they would not generally know what to do with it.

Impulsive donations have been relatively killed off because cash donations are banned (I think because scammers impersonate charities). So that leaves check and electronic payment. Oxfam does not (AFAIK) carry payment terminals. Checks would make sense, but they are taboo. So they have to ask for a bank transfer, which gives donors a chance to be lazy and forget about it.