Translation:Europe is not just one country, Europe is France, Denmark, Germany, England and many more!
Technically speaking America (on its own) is a country, North America and South America make up the Americas which are two continents :P
Why is 'meget mere' also correct (or at least suggested)? Aren't countries countable items, shouldn't only 'mange flere' be right?
I can't answer your question but translating the meaning of meget mere to english, it would be something more like much more
It depends on whether you're referring to the other countries (mange flere) or something more general, like culture or people, or just the remaining landmass if you like (meget mere). So both are okay.
'Meget mere' is not correct, it should be 'mange flere' as you suggest :)
The Danish sentence here doesn't refer to "many more countries", but "much more" in general. The given English translation is a bit off here and should say "much more". "Many more" would be "mange flere".