What anonymity?
eID is used by government services (which already know who you are because they issue your ID in the first place) and 3rd parties which rely on this identity. They only get access to data they need, you always have overview and control of it, right to be forgotten etc.
Yes but it enables services like facebook to do ID validation on everyone. It'll be much harder to stay anonymous because it removes a barrier to identifying oneself.
It starts with the government's own services but random third party operators are not ruled out, this is a fundamental building block for an "internet access pass" for all but the most niche sites like this one.
Right now Facebook only asks for ID in case of obvious fake names, because it's a huge barrier to ask people to send theirs just to create an account. This should remain a barrier.