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Which countries can I apply for a driving licence, without releasing mine?

Most countries won't take your old one. The exception are European countries where there is a general exchange treaty. This goes as far East as Armenia with all between.

For the rest, you need to check the treaty and the local rules. It is not very straight forward and often needs a phone call. Texas will take your licence while Louisiana does not.

The bigger question would be where you do not need to be resident. I think New Zealand, Egypt, Hong Kong and Australia would come to mind. Philippines is possible too but requires some trial or a visa excursion.
What if you move from one EU country to another. You have your driver's license from your home country, but you take the exam all over again in your new country, will that be possible?
 
What if you move from one EU country to another. You have your driver's license from your home country, but you take the exam all over again in your new country, will that be possible?
No, EU does only allow one driving licence per category. You cannot have the same one from two countries. This does not allow to Switzerland, you could do this. But honestly, it is a very costly endeavour. You better exchange it somewhere else where they do not take it and then go to Switzerland and exchange the other one.
 
It's not the same license from two countries though. What I wrote above indicated unique licences from two countries. But not sure if relevant if EU only allows one driving license.
Yes, you can take the exam in Belarus and keep you German one. But you cannot move to France and keep your German one as you are only allowed one for the whole EU:
https://europa.eu/youreurope/citize...ing-licence-exchange-recognition/index_en.htm
But not sure the point of it as you can easily convert it in Asia while keeping yours.