Really, it's not like insurance, because no insurer would structure their program the way that social security is structured.
But it is like insurance in that the program is named Old-Age, Survivors, and Disability Insurance (OASDI) Program.
It's also like insurance in that the payout is related to the premium / taxes. If you pay your premiums and experience the covered risks, you (or your survivors) get paid.
It's not like savings, because if you don't experience disability, or old age, you don't get paid. Your survivors might still get something though, I don't know much about survivor benefits.
Disability insurance is available from private insurers, but with different terms. Old-age insurance is more or less an anuity, again available from private insurers, with different terms.
The biggest difference with social security is that smaller incomes (and thus, smaller payments into oasdi) get a larger payment per dollar income if they experience a covered event. Another major difference is that if one has multiple former spouses of marriages that lasted at least 10-years, they're all potentially eligible for spousal benefits and they don't have to share it: no private insurer would sign up for that. Also, the actuarial tables are rarely updated and rather out of date at this point.