What an interesting question! It's generally assumed that The Princess Bride is a fairy-tale movie, particularly given the name, which alludes to an event common to many fairy tales: the wedding of a princess. However, in order to properly assess whether the film really is a fairy tale, we need to ascertain what a fairy tale is.
To begin with, a fairy tale is a fictional story, which The Princess Bride definitely is. Fairy tales also usually involve royalty, clearly defined heroes and villains, elements of magic, and a happy ending. In all of these ways, The Princess Bride adheres to the defined model. It also contains the common trope of a person being lifted out of poverty due to particular "good" traits and chosen to marry into royalty.
However, one major part of what makes a fairy tale a fairy tale is missing from this film. Most fairytales are folk tales, stories that have been told for centuries by parents to their children. The Princess Bride is not a story we have all heard many times before. Although it draws from the tradition of fairy tales, it is a new story in its own right.
We can also argue that "good" and "evil" are not as clearly defined in this story as we would expect from a traditional fairy tale. Inigo Montoya, in particular, is responsible for kidnapping our heroine, but he elicits the audience's sympathy to a certain extent because of the backstory he is given. Furthermore, the film does not end with the heroine happily married to her prince—on the contrary, she does not want to marry the prince; her love interest is a pirate, albeit a good one, and the wedding ceremony we do see is declared invalid. While The Princess Bride draws many elements from fairy tales, then, it also subverts these elements, toying with the audience's expectations to generate humor.