My old company ran social security, but I didn't get a social security card and didn't know my computer number. Now in a new company. Can I apply for social security again

If the two companies are in the same city, let the new company help you renew your payment in the Social Security Bureau. The original company should help you stop paying social security. If you are not in a city, you need a new company to open an account in the local social security bureau to pay for you, and then go to the original city to handle the social security transfer with the payment details pulled out from the new workplace.

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