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 objects, 匹 for small animals, 本 for long, thin things, 冊 for bound sheaves of paper, etc.
So as an example, an English interface could say something like "4 umbrellas", and the Japanese version could get away with something like "4 傘 [kasa, meaning 'umbrella']" if it's a list, but if it's in prose, you definitely need to be able to say "傘4本 [lit. umbrella 4-count-of-long-thin-object]" or something similar depending on the context.