Things get a little funky as far as the time and place goes in Beauty and the Beast. But, at the same time, they have a large bearing on how the story unfolds—and might even explain a few would-be plot holes in the bargain.
The one thing we know for sure: this is France, where the original story was written and which has always formed the background for the tale.
Jeanne-Marie Leprince de Beaumont penned the original fairy tale in the 1750s, which the filmmakers took as the basic cue for the costumes…and technology level. (Of course, they took their share of historical liberties in order to give it a more timeless "fairy tale" feeling.) We're left to speculate, however, on who the prince is, exactly, and how he fits in with the royal bloodline of France. His is a pretty self-contained kingdom, after all, and we never see any references to the people beyond the castle, the woods, and the nearby town. If he were more important, more people would notice when he turned into a mutant warthog, so we can probably assume that his kingdom is actually a very small part of France.
Or, you know, made up.
Why is that such a big deal? Disney clearly wants to keep the official roots of the story intact while continuing to give it a frothy "once upon a time" shine. So, while it's set in France and most of the main characters have French names, the story itself is more akin to other Disney films set in fictional kingdoms that have no political reality. It makes it a more universal story.