"wish" would be a perfectly acceptable rendering of this sentence into English. It should be an acceptable solution. Again, the rigidity of what is an acceptable English translation from the Italian is a continuing flaw in this program.
"to wish," just like an English, is a different Italian verb: «desiderare» → «Puoi stare qui se desideri.». Also, "if you would like" is a different mood, the conditional, of the italian verb «volere»: «vorrei».
There's just something not very gracious or polite about that. I think "you can stay here if you wish" is preferable and should also be accepted. Having said that, the "want" version would be OK among people who knew each other very well.
Firstly that sentence in itself would be wrong grammar. That would only be used if it were a question like (are you able to stay?). But even if it were to be correct, then I believe they could have used "Potete" which means "you can/ you're able"
I answered you may stay and was counted wrong. The "can" may be colloquial English, but the "may" is actually correct. I wish Duo would not correct my English when there is plenty of correction to be done on my Italian.
Yes, I agree with you. In some other sentences Duolingo does accept may. I think it's just a matter of DL over time getting better and accepting perfectly good, or in this case better, alternative translations.