Do you have a social security card for only paying pension insurance?

Yes, take Shanghai as an example.

The Shanghai Social Security Card Management Measures have corresponding regulations for it:

Article 11

(Application) A person who has a permanent household registration in the city and is at least 16 years old may apply for a social security card. Those who apply for a social security card shall go to the social security card application point in the street (township) where their hukou is located to go through the following procedures:

(1) fill in the registration form for application for a social security card in Shanghai;

(2) hand over their resident identity card and resident hukou book or certificate of household registration;

(3) go to the designated place to provide image information.

Expanded Information:

Related Laws and Regulations of Shanghai Municipal Measures for the Administration of Social Security Cards:

Article 13

One month before the expiration of the validity period of the social security card, the Municipal Information Service Center shall prompt the cardholder to apply for a replacement in a timely manner. The cardholder can apply for a replacement to the social security card application point in the street (township) where the household is located.

Social security card surface is defaced, mutilated can not be recognized, or can not read and write on the card-reading equipment, the cardholder can apply for a replacement to the street (township) where the household registration social security card application point.

Article 14

Social security card quality problems, by the Municipal Information Service Center is responsible for free replacement; belonging to the improper use of the cardholder caused by damage to the cardholder to bear the cost of replacement.

Sogou Wikipedia - "Shanghai Social Security Card Management Measures"