I agree with the OP that this terminology is awkward and confusing. Here's my take on it: native English speakers who are well-educated are used to terms such as "twofold" or "threefold", which sound just a bit antiquated but still usable. Critically, however, we don't see these words as containing the noun "fold"; "fold" is more of a suffix here, a funny special construction that is combined with a number to make a colorful variant on "double" or "triple", etc. It has absolutely nothing to do with the verb "to fold" or the noun "fold" that might come up while doing origami and referring to a folded piece of paper.
I suspect that the word "fold" started being used as a noun meaning "partition" in the context of k-fold cross validation when a speaker/writer not as familiar with English or with cross-validation thought that "k-fold" literally meant "making k 'folds' of the data". It's quite understandable that someone would come to this conclusion. However, "k-fold" doesn't mean "making k 'folds'" -- instead, it means "doing cross-validation k times", where the detail of having to also make k partitions of the data is implied.
Personally I never use "fold" in this strange way; I call the data segments in question "partitions", and it's much more clear.
Also, just because this usage has spread through the community doesn't make it reasonable English usage, IMO. I prefer straightforward and clear communication to inventing and using confusing new jargon.