Last year I tried eggnog for the first time, following a homemade recipe involving egg, cream and sugar. The result tasted like a thin vanilla ice-cream milkshake (with spices and alcohol). Recipes for homemade ice-cream use similar ingredients (without spices and alcohol).

Is this just superficial, or do the definitions of ice-cream milkshake and eggnog overlap so that it is reasonable to conceive of eggnog as a flavoured milkshake?

(I am looking for a technical answer that rules out my milkshake theory, or suggests it's an acceptable substitute.)