Can I take the medical insurance card passbook personally?

No, the medical insurance passbook must be collected by the company. You need to bring the company's social security registration certificate, a letter of introduction with official seal, and the original ID card of the recipient to get it. You ask the people in the unit to try it, or it may be from unit B. It's hard to say.

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