I would have said: Could you please make me another one? Is that correct?

Yes, this is a colloquial way of asking someone to make one for you. Grammatically, the 'can you make' is actually asking, 'are you capable of making', whereas what you are really asking is 'would you make me one too' - so in a grammar exam you need to know the difference, but in terms of day to day speech, your question is fine.