Private property in this context means things which generate/are used to generate capital, not just any kind of object which people might have and use. The important distinction is that capital is social, it is a means of coercing others to do work for you. That’s true for a factory, where people work for the owner, or for a rented property where the tenant must work to pay the owner. It’s true in a way even for wages - when you spend money you are buying the products of people’s labour (which under capitalism was not produced in a just way). It’s not the case for your toothbrush.

The distinction that liberalism made was that everyone should in theory be allowed to own private property rather than royals appointed by divine right and hereditary nobility they delegated some power to. Not that in the 1700s we were suddenly allowed to have our own clothes for the first time in history.

For leftists that’s exactly who we’re talking about when we say liberals. The right to private property and equality and the consent of the governed are logically incompatible. Right liberals (e.g. US Republican party) emphasise the former, and moderate liberals (e.g. US Democratic party) pay lip service to the latter while only actually protecting the former. It’s really only about property in the end.