Anonymity is of course weakened, but that is a different debate. Currently there is no anonymity at all if your account is linked to your email address, which you've also used to register your online banking account, your credit card verification, your PayPal account, etc. and when you have your mobile phone number confirmed, etc.

A password gives you 0 extra anonymity in this case. All it is used if for "identifycation" and in this regard it sucks in comparison to true biologial identification.