The problem with this is that every vehicle would need to be built around the same battery pack dimensions

There’s a lot of ways to tackle this issue. You could have a couple of standards (think AA vs AAA batteries). Or you could make the packs smaller and more modular so different applications can have more or less of them.

have the same amp-hour rating

No, they’d not need that. In fact, I’d say it’s desirable for them to not have that.

same voltage, same cooling system

Same voltage, yes, same cooling system? Not exactly. They’d just need to have cooling system hookups in the same place.

I seriously doubt that would ever happen as nothing like that has ever existed in the 120+ years of automotive history.

Loads of things like that have existed in the automotive industry. In fact, that’s one of the biggest features of the big 3 automotive manufacturers is having standardized parts shared between one another.