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).

Egg nog is a custard. The only difference is you don't freeze it in an ice cream mixer. It has all the same ingredients as ice cream except a heavier use of alcohol.

I haven't tried but I'd bet you could freeze it, too.

For one thing, eggnog (around in one form or another for hundreds of years) significantly predates milkshakes. Even switching the order of invention, though, still no :)

A milkshake is based on ice cream and milk, blended with flavoring. Some variants don't include ice cream, but a milkshake is always thick and cold. It doesn't include any eggs. You could have an eggnog flavored milkshake by adding spices and alcohol, but it wouldn't be eggnog.

Eggnog must include eggs, with cream and milk added to make it richer and thicker. Traditionally it is alcoholic (and milkshakes traditionally are not). While not served hot, it is never as cold a drink as a milkshake.

The similarity is not superficial in that the ingredients and preparation of eggnog is essentially the same as those for making a custard style ice cream. In effect, drinking eggnog is drinking unfrozen custard style ice cream base, with some liquor thrown in.

The difference is in the subsequent preparation: a milkshake involves blending ice cream and milk and flavorings into a frozen thick drinkable concoction while eggnog is traditionally served warm.

Is eggnog a milkshake? No the final product is too different. Is the similarity superficial? Again no, the ingredients and initial preparation are essentially the same (though the proportions will differ).

No. Eggnog is more a thin custard (especially the lightly cooked versions, the raw egg recipes less so)

A milkshake is made from milk, while eggnog is made from cream. If an eggnog has milk, it is technically a form of milkshake. However, this would be stretching the definition of milkshake. The same applies to the cocktail argument.