It’s plainchant, but the melodies clearly aren’t Gregorian.
Gregorian Chant is largely from the 10th to 13th century, and earlier in some cases. Missa de Angelis is from the 15th and 16th centuries and is more ornate and melismatic. Also Credo III which typically is used at the Mass of the Angels is from the 17th century. The exception appears to be the Sanctus for it, which is from the 11th and 12th centuries.
Some of the Kyries, such as XIa, have a simpler melody that is clearly from the Gregorian era, but also have a more melismatic variant that is much later, like XIb which is clearly an embellishment of the earlier melody, and is placed in the era from the 14th to 16th century. But at least it has a clearly obvious Gregorian root. Kyrie VIII from the Mass of the Angels doesn’t appear to have an earlier Gregorian root.
Another factor for the Mass of the Angels is that it is in mode V, which is very rare in the Kyriale, and only used for the Kyrie and Gloria with Mass of the Angels as far as I can tell, except for one ad libitum Kyrie.
Overall while I think it’s nice to hear from time to time, it’s overdone. Here are some other nice ones, that
are Gregorian chant:
Missa Cum Iubilo (IX) for feasts and solemnities of Our Lady
Missa Orbis Factor (XIb) for Sundays in O.T. (after Epiphany and Pentecost pre-Vatican II)
Kyrie Lux et origo (I) for paschal time
Gloria I for Paschal time
Gloria XIII Stelliferi conditor orbis, ad libitum for Sundays in Ordinary times I find this one particularly haunting.
I’ve just scratched the surface
Here’s a bit more on the overuse of Missa de Angelis:
newliturgicalmovement.org/2007/12/musical-rubrics-and-missa-de-angelis.html