with is "con", so even if it sounds better in english, that doesn't mean it's right for the translation.
"He is popular with the students" is probably more idiomatic English than "He is popular among the students".
I actually disagree. If you think about it "with the students" wouldn't grammatically make sense. Are you trying to say that the students are also popular? He is popular among the students is the best grammatical (not cultural) answer.
In another item, "Eres popular entre los niños" (pardon the lack of accents) suggested the answer "You are popular WITH the children.", but using "with" instead of "among" to translate "entre" here is marked as wrong. What gives?
In English, to me, being popular WITH the students imples that the popular one could be a student or more probably some other class of person. Being popular AMONG the students would more likely imply that the popular one is a student himself/herself. That's how I would use use the words.
In Spanish, I don't know if such as distinction would be made betwee ENTRE and CON.