People owning stuff is personal property.
The aim is not to abolish personal property. Individual consumer goods would be personally owned. Private property has the technical connotation of ownership of the social productive means that are necessary to production in a society with an enormous division of labor or soicalization and specialization of the production process. The fuller statement of the goal is abolition of private property in the basic means of production, for which abolition of private property is shorthand. So, individuals would own cars, but not auto manufacturing enterprises.