A number without a measure word doesn't make sense in chinese, every noun quantity that is stated in mandarin needs a measure word after it. The only exception is when you say "3 days" or "3 years", because days, years and some other quantities are measure words themselves.
The most common measure word is 個 (个) and if you don't know the particular measure word you can always use it.
One difference in chinese is that the specifier (This, that, which) will have a measure following after, so you see 個 all the time, but you don't see that with japanese specifiers (sono,kono,...) . Both definite and indefinite objects have a measure word following after in mandarin.
The measure word can tell you what kind of object you are dealing with, like there's one for animals, thin objects, glasses/cups, books,.... Works the same way in japanese i think, but sometimes its not as obvious in japanese because 一杯 which means 1 glass is written in hiragana いっぱい, but i have noticed 本 used for books quite often.
So to answer you question i would say there's probably the same amount of measure words in each language, but chinese uses them more because of the specifiers. 個 is the default measure in chinese so you see it all the time but im guessing it's more specific in japanese
if you're interested i can tell you're more about mandarin