Can a company, as a legal person, pay social security by itself?

You can't. Because a legal person is not a natural person, it is an organization that independently enjoys civil rights and undertakes civil obligations according to law, and cannot enjoy civil rights and undertake civil obligations in its own name, so an enterprise legal person cannot pay social security on its own. A legal person is an organization that has the capacity for civil rights and civil conduct, enjoys civil rights and undertakes civil obligations independently according to law. It can be a company, not an individual.