Mawile is based on [Futakuchi-onna], a Japanese myth about spirits of women who possess two mouths, one normal and one large fanged mouth coming from the back of the head. This relates to the Fairy-type because it relates with supernatural, metaphysical or surreal beings.
As for Mr. Mime, this is a trickier one that relies more on assumptions and deductions. Aside from obviously being a mime or clown, it also bears resemblances to marionette dolls. Mimes also tend to be tricksters, something that could be a connection with traditional fairies. Not Disney fairies, but fairies from European folklore that were known to be mischievous and even malevolent. Mr. Mime takes pleasure in watching people walk into invisible walls it makes. There is also the fact that it's pre-evo, Mime Jr., resembles a gnome or small troll, which also tap into the supernatural realm which the Fairy-type hails from.
Here's some stuff I found on wikipedia about traditional fairies.
>Much of the folklore about fairies revolves around protection from their malice.
>Some pranks ascribed to them, such as tangling the hair of sleepers into "Elf-locks", stealing small items or leading a traveler astray, are generally harmless. But far more dangerous behaviors were also attributed to fairies. Any form of sudden death might stem from a fairy kidnapping, with the apparent corpse being a wooden stand-in with the appearance of the kidnapped person.
>A considerable amount of lore about fairies revolves around changelings, fairy children left in the place of stolen human babies.
About that last one… it's not that hard to imagine Mr. Mime stealing human babies.