"I have 80 cents."
It would be okay if the topic of the conversation was already about money, but could be confusing if you just make a statement like that with no context.
毛 also means hairs so it could possibly be interpreted as "I have 8 hairs"
Nope. That'll be more like saying I have 8 fur points which hardly makes any sense. And if you meant 八美分, you just said that you have 8 cents instead of 80
Unfortunately thst makes no more sense than '八块毛', or in English, a phrase like '5 dollars cents'.
We're mostly used to currency that has two units: the main unit, and a hundredth of that unit. In Chinese currency there are three, but just like dollars and cents they are different units, and it only makes sense to say the name of the unit if there are some.
Chinese currency is divisible by 10, not by 100 like say Canadian Dollars. If you go into a store you will never see anything marked as say, 9.99 because there is no division beyond the 10s; instead you would see 9.9. It's wrong to call 毛 dimes because that implies further division is possible.
In the context of money, 毛 means 0.1 yuan. But otherwise it means fur, hair etc!
You'll notice that the questions that require 毛钱 and not just 毛 are those that don't have another money unit like 块 in the mix.
It might be mathematically correct to say 80分, but that isn’t the way currency is expressed.
Note also that 分 has many uses, so you would need to include 錢.
I would also add, for the same reason that you would say "it costs 8 dollars" and not "it costs 800 cents" despite them being identical, mathematically speaking.
According to the usual monetary order, cent is the unit directly under dollar, like 毛 under 块, but according to value, 100 cent = 1 dollar, then cent would be 分 because 1 块 = 100 分.