Directly translated "the nose of my dog is very big" should also be correct.
It may be grammatically correct, but it sounds awkward - as if someone who is not a native speaker of English came up with that phrasing. Plus the course contributors may be trying to make a point about the different grammatical structures Spanish and English typically use for possessives.
It does sound awkward, but as a relative beginner, it's enough of a challenge for me to just translate the words correctly. I still think my answer should have been accepted.
It seems like "the nose of my dog is very large" should also be accepted.