Earlier quoted context omitted.
English has some of these words too -- they're specific kinds of counters. Paper is counted in "sheets" or "reams," for example. "I want a sheet of paper" is distinct from "I want a paper" -- outside of the context of a newspaper, the latter would be considered grammatically incorrect. It's just that in Japanese, these are more or less standardized based on the kind of thing you're counting, so 個 for small, discrete…
Upthread, the topic is set by yorwba's comment: >>> Japanese doesn't just have different measure words, but also different number words depending on the thing being counted. You're talking about the measure words, not the number words.
I understand that; I mistakenly thought mrunkei was unclear about measure words.