The music is composed to give the impression of "magical" and "fairytale"-ish feeling, usually involving the use of piano and choir. The scene itself usually portrays a "medieval" (so to speak) or fantasy setting.
I can't describe this better, so here are a few examples,
Where and when did this kind of music originate? Does it have a Middle Age roots? As far as I know medieval music is very monophonic and didn't involve too many musical instruments, so I suspect this might be modern invention, but of course this is an uneducated guess...
The music you give as an example sounds like it is influenced a lot by impressionism.
Improssionism is a musical style from the late 19th/early 20th century and the two most important impressionist composers are Maurice Ravel and Claude Debussy.
Prèlude à l'après-midi d'un faune(Prelude to an afternoon with a faun)
Introduction and allegro for harp, flute, clarinet and string quartrt
You probably have noticed that in cartoons if a character falls asleep they play a certain dreamy scale up and down. This is called the whole tone scale. The first part of book 2 of Debussy's images uses this for example and it sounds very dreamy and fairytale like.
You also mention the use of piano and choir. Debussy himself was a very prominent composer for piano and a virtuoso himself. Ravel also has written a lot of music for piano. (Warning speculation ahead) I assume that this is the reason of the prominence of the piano in the use of impressionistic elements in film music because the impressionistic harmonies are already associated with that kind of instrumentation.
The same thing goes for the use of the choir. Debussy used a wordless choir in his nocturnes for orchestra(links below) and Ravel used it in his ballet Daphnis et Chloe. Another prominent impressionist composer for choir is Lily Boulanger.
Nocturnes Part III Sirènes - Debussy
Veille Priére Bouddhique - Lily Boulanger
Does it have a Middle Age roots?
Not in the way you might think. An aspect of impressionistic music is that it can be modal and the use of modal scales comes from the renaissance era. I do not know how much technical knowledge you have about music theory so if you want me to elaborate, please ask!
I hope this helps and I hope I didn't get too carried away:')
I hope some other people might add to this post by talking about the music of Rymsky-Korsakov's, Holst's and Tchaikovsky's influence on this type of music. I know that they had a big influence on this type of film music as well but my knowledge about them is severely lacking.