It could mean many things. A milkshake comes in many flavors, is cold, is sweet and makes a person feel good. I would interpret as that the sky might be sunny (a vanilla shake) or simply to soak up (drink) the feelings the sky is giving. Let the sky fill you with feelings of happiness, of friendship or making you feel good like sharing a milkshake with someone special can (:

I have never heard that expression used in the English language. I would think it might mean a cloudy (white) sky.

If it's a poem or a song lyric, it's just some feel-good words that sound nice, and vaguely make sense.