"He has a lot of coats."
Translation:Er hat viele Mäntel.
Because you always mean multiple coats when using viele and same applies to viel. Viele is in pluralis and gets therefore the ending -e.
Vielen is not probably used in German, because it refers to a singular masculine object. There can never be many of one object, the reason why vielen is not used.
ex. "There are many man" would not make much sense, whereas "there are many men" would.
This is arguable. I think, there is enough difference between a "Mantel" (coat) and a "Jacke" (jacket) not to accept it.
In my opinion it is better to be somewhat strict with the acceptance of answers for the following reason:
If you choose a word that is considered wrong by Duolingo and get corrected, but the word is not very wrong, you can decide for yourself, whether you think you have translated the sentence well enough.
But if too many translations are accepted, you might not always know whether your translation is a good one or just a merely acceptable one. It starts to get somewhat blurry and it is harder to learn the right translation for the words.