You’re a bit confused about the Marxist notion of the State, understandably if you aren’t a Marxist. For Marx, the state is the aspect of government that entrenches and enforces class distinctions, ergo once all property is public there are no classes, and thus no state, despite a government remaining. Per Engels:

The first act in which the state really comes forward as the representative of the whole of society – the taking possession of the means of production in the name of society – is at the same time its last independent act as a state. The interference of the state power in social relations becomes superfluous in one sphere after another, and then dies away of itself. The government of persons is replaced by the administration of things and the direction of the processes of production. The state is not “abolished”, it withers away. It is by this that one must evaluate the phrase “a free people’s state” with respect both to its temporary agitational justification and to its ultimate scientific inadequacy, and it is by this that we must also evaluate the demand of the so-called anarchists that the state should be abolished overnight.

Additionally, money can only be abolished once an economy has fully socialized, at no point in the USSR’s history was that feasible. They even tried to move to a labor voucher system, but lacked the computerized means to make it truly practical.