It depends on where you are. In parts of the northeast USS, this would automatically be disqualified as a milkshake because it has ice cream in it. Around these parts, a milkshake is, believe it or not, shaken milk with flavoring.

If it has ice cream, it's a frappe in Massachusetts, and a cabinet in Rhode Island (where a coffee cabinet is in fact a drink). Maybe you could invent your own term!