That's an interesting question. Here's what I think.

I don't think it shares any melodic material with either of the songs you mentioned -- at least, with the versions I've heard (I'm sure there are ones I haven't). The air sung by both Sorcha Ní Ghuairim and Máire Uí Chéide is the same. The first and last lines of the stanza are sung to tone melodic phrase, and the two lines in the middle are sung to a second melodic phrase, giving an overall ABBA structure. By contrast, Amhrán na hEascainne (the Connacht version of the Lord Randal song) is melodically structured AABA, Cá rabhais ar feadh an lae uaim (the Munster version) is ABCD, and Eileanór na Rún is sung to an elongated and rather complex air befitting the complexity of the song's metrical structure.

However, what may have struck you is the clear metrical similarity in these three songs. (Note that this applies to Eileanór na Rún only as far as the first three lines and does not apply to the repeated lines beginning "a bhruinnilín deas óg...") Here's a stanza from each one; I have divided line 3 into two phrases where appropriate, and indicated assonating vowels in capitals in the first song; the others have the same structure:

1. Maile Ní Maoileoin

Thug mé liom isteach sa ngairdín í mo chuid den tsaol is mo STÓR,
Is rug mé i ngreim barr láimhe uirthi is bhain mé di dhá PHÓG,
Níor lig mé as an ÁIT sin í
gur bhain mé di mo SHÁSamh,
Is ansin is ea rinneas FEALL ar phlúr na mban ÓG.

2. Eileanór na rún

Bhí bua aice go mheallfadh sí na héanlaith ón gcrann
Bhí bua eile aice go dtógfadh sí an corp fuar ón mbás
Bhí bua eile aice nach ndéarfad,
Sí grá mo chroi is mo chéad searc.
A bhruinnilín deas óg
Is tú is deise milse póg
Chúns 'mhairfead beo beidh gean a'm ort
Mar is deas mar a sheolfainn gamhna leat, a Eileanóir na Rún.

3. Amhrán na hEascainne (Lord Randal, Connacht version)

Céard a fhágfaidh tú ag do mháithrin, a dhriotháirín ó?
Céard a fhágfaidh tú ag do mháithrin, a phlúir na bhfear óg?
Dá bhfágfainn saol brách aice
D'fhágfainn croí cráite aige
Mar tá mé tinn fá mo chroí agus béad go deo deo.

If you wanted to, you could sing Maile Ní Mhaoileoin to the air of Amhrán na hEascainne -- it would work pretty well. Eileanór na Rún is another matter on account of the extra text. Oddly enough, the Munster version of the Lord Randal song does not have the same metrical structure as the Connacht version.

I know this isn't the appropriate place for self-advertisement, but as it happens I have just published a book dealing with poetic metre in traditional Irish verse. It's called "Irish Song-Craft and Metrical Practice Since 1600," and was published by the Edwin Mellen Press in January of this year. If you want to know more about it -- it's meant to be helpful with questions like this -- you can go to the Mellen Press website and search under my name. Unfortunately it's very expensive; maybe a library would be willing to order it for you. BTW, I will not be receiving any royalties, so there's nothing in it for me except fame (or possibly notoriety)!

Virginia