It's not the government directly, but what is called in German "Flucht ins Privatrecht" [escape into private law], meaning that the government "outsources" such activities to private organizations that are only very indirectly charged by the government (implying that you cannot use public law to sue the government, but you have to sue the respective organization indirectly. Also, since the relationship between the government and the respective organization is very indirect, the politicians can claim that they are not responsible for the organization's wrongdoings - something that is often not easy to disproof).
From https://cuii.info/en/about-us/
The CUII was founded by Internet access providers and rightholders and coordinates the implementation of court blocking procedures and the enforcement of court blocking orders.