[intertextuality]

The way they function is completely different.

English: measure certain mass nouns or classify certain nouns. Not used for counting every noun.

Korean: Always used, regardless if they measure or clarify. Six dogs, three months, two papers, four volumes, one bowl, ten things all use counters. Always.

The difference is night and day.

[thaumasiotes]

The fact that they are used for all nouns [1] in Chinese / Korean and only one of the two classes of nouns in English is not a difference in the way they are used or the function they perform. It is a difference in which nouns require their use. They have identical grammar and meaning. There is no reason to consider them distinct phenomena; they are one and the same.

[1] Certain things that you might think of as a noun based on English are, in Chinese, measure words which do not measure a noun. For example, 天 "day" and 次 "time" (as in "it happened three times") are syntactically measure words, but it is not possible to follow them with a noun that they notionally measure. This isn't really distinguishable from saying that all English count nouns are really measure words.