What goes around comes around. There is nothing new under the sun, etc. It's certainly not Gregorian chant, a very specific form. What whoever told you that probably meant is that like Gergorian chant, the Alainu tune is plainsong, a free-flowing form that lacks the regular metrical structure that most Western music has had from the 17th century on. Most hazzan chants are plainsong, at least in the Ashkenazic tradition.
Recordings are a bit hard to find, but it's fascinating stuff.
However, I have to admit that by ear I find it far-fetched, as it sounds to be in a Wester major key, and has Western cadence resolutions and the like. It's not at all unlikely that this is a massaged, Westernized melody with sources in Chant, though. I'll hang on, and if no one can confirm by about Nov. 5, I'll go tickle a source. Thanks to MJ we now know that the origin of the Aleinu tune comes from a children's song. I once heard that Anim Zemirot was also a children's tune, but I can't seem to remeber which song it was.