I have recently taken few lesson of Gregorian Chant and we sung something from the Ninth Mass to Virgin Mary. Unfortunately, we stopped at Kyrie and I've heard its Sanctus just once or twice, so I can't tell whether some part of it is similar to that Aleinu record or not. I have notes for the Sanctus, so at the first opportunity I can scan the notes and link the picture, so that someone more competent can verify this theory. The guy who taught me the chant has been to France few times, so it should be what we are looking for, not "central-europian mutation" of the same chant. Unfortunately, I'm not really competent in reading (medieval) musical notation, so there might be some piece similar to this even though I didn't find any similarity.
Anyway, even if the two melodies were somewhat similar, still it is quite likely to be just an accident. There is limited (though big) set of possible melodies, and set of melodies used for sacral music is not-as-big. Provided there are some similarities between Jewish sacral music and Gregorian Chant, such occasional similarities are inevitable.
Such random similarities in music are often exaggerated by conspiracy theorist. In my nation (Czech) vague similarities between our and Irish folk music are often presented as a proof that we are more Celtic than Slavic, but I remember one of the proponents of this theory to admit that he have heard music from Tibet that sounded more similar to some kinds of Czech folk music than Irish music does.