You are absolutely right.
The phrase "Μπορούμε να του δώσει το κρασί" is incorrect. The correct one would be "Μπορούμε να του δώσουμε το κρασί"?
The correct translations are: "Can we give him the wine"? or "May we give him the wine"?
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.