That is not fully true. At least in East Germany you owned things. You could own a car and the furniture in your house.
Soviet doctrine (and the broader Marxist doctrine) distinguishes between "personal property" and "private property". Things like furniture or car would be considered personal property, and hence okay. Land, means of (large-scale) production like workshops and factories etc, would be considered private property if owned, and that was banned. Houses and other things that straddled the line could be treated differently depending on the country and the era.