It does not mean that usage is monitored, and usage is the important privacy aspect.
Furthermore, this just puts mobile phones on par with landlines. It used to be that having a phone meant having a landline at home, and obviously that meant knowing exactly who a number belonged to.
There needs to be a balance for everything in the real world. And even most Western countries have concluded that requiring IDs for SIM cards is a necessary measure.
Now, for Chinese authorities one issue (without even going into politics) is that China is a very low trust society where it is almost expected that everyone will flaunt the rules at the first opportunity. So I am half guessing that the government sees technological solutions as a way around that.
(As for renting a room, it would be crazy for a landlord not to know who their tenant is just for practical reasons considering land and tenancy laws in most EU countries)