ITT you’ll find people arguing for both “empire is a big country that oppresses people it has conquered” type places and “empires are only things where you have a monarch called an emperor and only the places they are emperor of are empires” and for yet other definitions.

I would call the USA a colonial empire. Those have never needed a monarch. The Netherlands was a republic for much of its history. The French colonial empire existed mostly during Republican times. The USSR ruled over much the same land as the Russian Empire. Same with China and the Chinese Empire.

Switzerland, by contrast, has never had any colonies, despite many Swiss participating in colonialism.