"He has a cute face."
“ 他的脸很可爱” literally means "His face is cute." Is it not possible to say "他有可爱的脸“?
That's possible. The literal translation is 他有(一张)可爱的脸. Quantifier: 张 zhāng
Duolingo can be so very picky about matching word order sometimes, and then it gives us this one, where the English and Chinese sentences don't really match semantically... this is what I get for trying to work without the word bank