The answer of this question explains that it is to install apps bought on the Mac App Store on multiple computers.
My problem is a little bit different. I moved to another country and I had to change the country also in my iTunes profile, due to my new credit card. As a result my iTunes store, App Store and Mac App Store are now associated to the new country.
How can I access and download my purchased Apps on my new MacBook Pro?
Probably one solution would be to change again the country, but it wouldn't be optimal.
Best Answer
For App Store, see this answer for "Do I need to repurchase apps if I moved countries?":
For MAS - recover purchased apps on Mac App Store (formatted better and corrected typos):
Also see iOS: Changing the signed-in iTunes Store Apple ID account