you cannot. A Kindle can only be registered to one account at a time. You have some options:
migrate one account to the website of the other and link them using Family Library. If you live in France I recommend using amazon.fr as they have pretty much the same books in English as amazon.co.uk plus they don't have the 20% VAT that amazon.co.uk puts on books. Then continue using amazon.fr from now on. If you live in the UK, they have almost as many books in French as amazon.fr but I'm not sure how Brexit will affect this in the future.
multiple devices: one registered to each site. I did this when I lived in France then I migrated my account to amazon.com when I moved back to the US and linked the two accounts.