I suspect that アシュリン (A-shu-ring) is meant to be
Aisling. Why? Because that's how my Japanese mother-in-law pronounces my daughter's name.
The lengthening has been introduced to fit the rhythm of the song and the final ン (ng) is very weak in Japanese, hence kk heard it as A shū (r)ii.
Another hint for that is that
aisling can mean "dream" and the title of the song is Yume no Owari "the End of a Dream" (or "Dreams" - Japanese doesn't normally bother with a singular-plural distinction, let alone definite or indefinite articles.)
My guess is that the songwriter wrote the lyrics in Japanese, then got an Irish translation (from Mr Google perhaps) and then just pronounced it as if it were romaji, i.e. Latin-based spelling with no distinction between l and r or b and v.