Also France and Germany are quite multicultural, so I don't see how it's any different becoming French or German vs American or Brit in general. It just comes down to you being Indian and probably only country having large Indian population in Europe being UK, so they can think you was already born there and consider you local. Black people could already pass also for French, Turks for Germans, but yeah specifically Indians will stand out everywhere except UK as not being born there, but just because you stand out it doesn't make your life worse.

Other people stand out for other reasons than color of their skin anyway even if they are completely local and still live in their countries without moving to country where people like them stand out less. By your logic very tall people should move to Netherland, dwarfs should move to some dwarf colonies etc. so they won't stand out for being different, your skin color is just another thing to stand out.