Can a person have two social security accounts?

There is only one account, and this is not absolute.

Social security is managed as a whole, that is to say, it is not allowed to open two social security accounts in the same overall planning area.

The crux of the problem is that if the new unit and the original unit belong to the same overall plan, it is necessary to transfer the social security relationship to renew the insurance.

Otherwise, you can immediately re-open an account in the new unit to pay social security.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.