Not really. If they didn't feel the need to use his actual name, I don't see why he needs an official name.

If they don't say his name in the movie, they don't need to name him. I've made movies before where we went to a lot of trouble to name characters, but then we never said their name in the movie

Well he was just a minor character after all. And that's really all we need to know about him. He's The Prince,it's cut and dry.

He was originally going to have a much larger role in the story but had to reduce his screen time because the animators found it too difficult to draw a guy (funny, how they're all men). So they just stuck with Snow White, the Queen, and the Huntsman as the only humans they had to rotoscope.

Maybe they just have no idea to give name to the Prince. Or because he was minor character in the film so his name isn't really important.

It's rather understandable though, he shows up twice during the entire film, and I gess there was realy no reason to give him a name at the time, perhaps? Or no appropriate moment to mention it? Snow White couldn't have known his name, and there'd be no reason to say it.

At that time, people don't really care about names. Not just the prince, but also the kingdom, the villain, or the parents. If you see Disney movies in 21st Century, they all have names, parents, and childhood backgrounds.

It gives a genuine fairy tale feel to me. In folklore, characters are rarely named. They are usually identified by a "title", like "the maiden", "the king's son" or "the witch".

That's how it was in the Fairytales. It makes it look more fairytale-ish. Besides the Prince was not all that important.

Most characters in the original fairy tales (apart from the titular character like "Snow White" or "Rapunzel") didn't have names either. This was back when Disney was TRYING to be faithful to the source material (they even wanted to include the poison comb and tightened corset, but ran out of time/money), and it gives a very dreamy, fairy tale feeling.