Does China Financial Social Security Card charge management fees?

No need. No matter where you apply for a card, as long as it is linked to social security and provident fund, you don't need to charge an annual fee, because you can apply for a social security card in a bank, which means that banks are licensed to cooperate with the Social Security Bureau, and not all banks can apply for social security cards and provident fund cards.

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