Is eggnog just a milkshake?
Not really.

For a start there's no milk in it (there's cream, but milk is the defining factor in a milkshake). Second, egg isn't a normal ingredient in a milkshake, and neither is alcohol. Of course they can be added, but they take you away from what's normally meant by the term. When that happens it's normally reflected in the name.

In general, trying to precisely categorise foods is an exercise in futility - defining a category that includes everything you'd expect it to invariably includes things that don't belong, or you end up with categories so broad as to be useless. 
Is a hot dog a sandwich? 

Even calling eggnog a cocktail (An alcoholic drink consisting of a spirit or spirits mixed with other ingredients, such as fruit juice or cream.) is slightly problematic, as "cocktail" usually implies it's mixed just before serving, rather than bottled in between (as is common with eggnog).