I think the main argument for having the government do it as opposed to the private sector is that the gov has a lot more restrictions and we, the people, have a say. At least theoretically.

Imagine if Walmart implemented an identity service and it really took off and everyone used it. Then, imagine they ban you because you tweeted that Walmart sucks. Now you can't get a rental car, can't watch TV, maybe can't even get a job. A violation of the first amendment in practice, but no such amendment exists for Walmart.