I usually use "can" regardless, because I have been marked wrong before for using "may" even though it was the better grammatical English. If I remember correctly, I reported one instance of it that was eventually changed. But it seems they will only change the exercise that you report.
You are correct. Even though 'another' is the literal translation, 'different' is even more appropriate as far as meaning. The speaker isn't asking to select a soap again in addition to the one they already selected. That makes no sense. They mean a soap different from the one in focus.
In cases where there are ambiguous translations the closest to the actual meaning should count. When we speak we are attempting to convey information. Calling the translation closer to the intended meaning 'incorrect' is missing the entire point of communication.
Maybe the people have been offered their choice among a basket of beautiful handmade soaps. They desperately want more than one of them--they're just that beautiful--but they finally manage to pick one. But then their desire overcomes them, and the ask the giver if maybe he or she might be willing to part with just ONE more of those magnificent soaps. Could happen! ;)
I wrote 'May we choose another soap' which is, in English, a perfectly acceptable translation of possere. Indeed, it's a more accurate translation because we are asking permission to choose, whereas to say 'can we' asks the question whether it's actually possible to choose!