What's the difference between social security in Xinzhou District of Wuhan and the city?

Yes, you can do it in Wuhan Social Security Bureau and any social security institution in seven central cities. If you pay it yourself, you can do it at the flexible employment window and open a new account to pay.

Xinzhou District belongs to the far urban area, so it can't be merged with the social security in the central urban area for the time being, and it needs to be opened separately in the urban area.

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