'Ye' is generally used as a plural 'you' but in the song lyrics it appears to only refer to one person (Danny). In that respect it's both inconsistent and illogical, but this is a song we're referring to.
As a side note, there is a type of ballad in traditional Irish music which is often referred to as a 'come-all-ye' (not hugely dissimilar to Bob Dylan's The Times They Are A Changin') which may have influenced the lyrics of this song.
"Ye" is the original second person plural nominative while "you" is the original objective case form of "ye"; so, it was originally Ye see me (Old English: Ge seoþ me(c))
and I see you(Old English: Ic seoeow(ic))
In early modern English the nominative "ye" was replaced by "you". In dialectal and poetic use the old form is still sometimes used.
Maybe because "come ye" is a set phrase the old form is preserved here?
Ye, in addition to its plural meaning, was also "sed instead of thou in addressing a single person (originally as a mark of respect or deference, later generally...)" [OED]. The OED has many examples which demonstrate, over the course of four or five hundred years, the change from it being a mark of respect to its general use in respect to any person. The later examples seem to show preservation in dialect and regional speech rather wider usage.