Well, not neccessarily, this sentence could be said with «мне» too. I think both variants are equally OK.
Watching tv with a Russian friend, i saw a scene where a bride at dinner sweetly says "дай мне вананчик" to her new husband. It seemed so blunt but my friend said the diminutive 'чик' meant it was polite, even without a 'please'. It seemed odd. And still does!
Also odd that you can add an affectionate diminutive to a random noun too, but that's for another conversation i guess!