We have private tax-advantaged plans that work essentially like that, but they are totally optional. Interestingly the only system that is mandatory is our “Social Security” - you are obligated to pay in but the money is pooled; you get something akin to shares in the system, but it only entitles you to a specific payout after a certain age; your contributions are not privately owned.

Unusually for these types of discussions, it seems like the US system is more socialized than the alternative in your country.