No, đĩa literally means a plate/dish. That's the Northern term, the Southern term is dĩa. Here are some differences between North and South:
English - North - South
Dish/plate - đĩa - dĩa
Fork - dỉa - nỉa
Spoon - thìa - muỗng
Bowl (big) - bát (lớn) - tô
Bowl (small) - bát (nhỏ) - chén
Glass - cốc - ly
Grammatically speaking, yes đĩa is a classifier. It is used here the same way 'con' is used for animals (con cá, con gà, con cái etc) in the sense that it is needed in order to quantify (ie count) something.
That's all that classifiers really do grammatically in languages that have them: they're used to count or specify quantity.
I think the answers you got misinterpreted your question, while this sentence does imply a literal plate, it implies it as a unit of rice as you noted - not a plate made of rice. You could in theory say "Họ ăn một bát cơm" or "Họ ăn một muỗng cơm" (thanks @ TehVanarch for the handy translations for 'bowl' and 'spoon') to get across similar meanings.
Source: I'm a linguist and I know some Mandarin Chinese, which also uses classifiers. I'm just a beginner in Vietnamese, though so take this with a grain of salt lmao
I have had some exposure to Chinese and Swahili, so I have some sense of what a classifier is. That said, I have always found classifiers like a "plate" of rice or a "sheet" of paper a little confusing, since it has more meaning than just the classification. "Con," for instance, doesn't seem to add anything other than than it is an animal--one animal of rabbit, five animals of cat, etc. Rice, though, could be in a bowl or a pile or a handful, so I wasn't sure whether, for instance, one would still use this classifier, if it were in those vessels. If I were handing you two kilos of paper, for instance, would I say two kilo sheet paper or just two kilo paper, with kilo being a classifier? In other words, I am not sure how different they are from the various units we use in language that we do not think of as having classifiers. Perhaps learning the classifier "mon" for these animals as meat in the food lesson will begin to clear this up for me. In any case, thanks for the help.