It's fundamentally different in any society with strong private property rights. A private pension is a set of investments, i.e. something that ultimately is owned by you. How much those investments pay off, if at all, is unknown, but if they do pay off then it's yours.

A state pension would be illegal for anyone except the state to run, because it'd be classed as a Ponzi scheme. Those are recognized as being unstable, hence why you're not allowed to run one. There is no investment into owned assets that pay out, just the appearance of it.