This probably allows them to maximize profits, e.g. most French people probably won't go look at Amazon Spain to see if the graphics card they want is cheaper there...
You are correct in that Amazon EU has common stock despite the products having different prices on the different regional sites.
Price discrimination would be a reason to do this but then it's strange that they'll ship from every website to me and have a single login. They've even recently allowed changing the language on a few of the websites. But maybe they're just changing the prices per-user in the US and can't do that in the EU so they do it per site? Whatever it is it results in a really bad user experience.
This probably allows them to maximize profits, e.g. most French people probably won't go look at Amazon Spain to see if the graphics card they want is cheaper there...
You are correct in that Amazon EU has common stock despite the products having different prices on the different regional sites.