Great movie! Easily makes my top 5 favorite of all time. I would definitely say it has more in common with fairy tales than not.
For example, the film has a clear division of good vs. evil. It's clear that Wesley is good and trying to do good, and it's clear that the six fingered man is trying to do evil. Fairy tales also tend to focus on royalty in some way. That is true of the film as well. Buttercup is a princess, and she is in a situation where a royal wedding is going to take place. I would say the most obvious fairy tale element within the film is the presence of magic and mysticism. That's the main function of Miracle Max and his wife. The story also ends with the standard happily ever after ending.
The parts of the film that are not like standard fairy tales are fewer. One example is that the hero is wearing black. That color motif is generally reserved for identifying the main "bad guy." While all fairy tales have some sort of conflict, The Princess Bride is definitely darker in its tone. Sure the film is light hearted, but there is a lot of violence and emphasis on killing. That type of thing exists in other fairy tales, but not to the extent that it is exists in the movie. I would also make the argument that Wesley isn't as "pure" a hero character as other fairy tale heroes. He is the current Pirate Roberts, but he is not the original. He has assumed the title and will pass it down to another. In essence, Wesley is knowingly perpetuating a lie.