Private property used by marxist philosophers refers to property that generates capital. An example would be a factory.
When marx said abolish private property, what he was really saying is, make it so that factories are owned by the people who work in them, rather than by some rando who has nothing to do with working in them. He was not saying that you shouldn’t have the right to own a toothbrush.
Your toothbrush, according to marx, would be PERSONAL property.