A while ago I made an outline of what makes a song magical for me. Eventually I will flesh it out into an essay sometime. It has three departments: Music, Words, and Both. Both comes first.

BOTH
>Long acquaintance
>Power (old songs only)
    Abide with Me
    National anthems
    Internationale
    Horst Wessel Lied
    Marching through Georgia
>Variation, repetition, recollection
    Lord Randall
    Dillan Bay
    Love's Old Sweet Song
    Where Have All the Flowers Gone?

MUSIC
>Harmonic interest (opportunity for showy chord progressions)
>Melodic interest
    A new note near the end, especially if out of the previous range:
      Banks of Marble
      Internationale
      Horst Wessel Lied
>Rhythmic interest

WORDS
>Conviviality
>Consolation
    Cf. Power
    The palpably untrue:
      Johnny Macree
      The Parting Glass
      hymns
>Exotica
    Time & place: Eppie Morrie
>Malice
    The characters'
    The narrator's:
      The Parting Glass
    The author's
    The singer's
      Thru identification
      Thru offense given
    _Not_ the composer's
>Particularity, vividness
>Poetry
    Prosodic agility
      Simultaneous alliteration & rhyme: Married Girl
    Linguistic daring; allusion; wit
>Puzzles (Cf. Exotica)
    Schnitzelbank
    Miss Bailey
>Sentimentality
    The outside: death, love, marriage, misery, poverty, conflict, pain

Most of the songs that I find magical happen to be folk songs -- or at least, of anonymous composition. That, however, is merely an odd fact, not a requirement.