No it doesn't.

It's impossible for ordinary users to distinguish between privacy invading tracking cookies and regular functional site cookies.

Also, this doesn't form "informed consent". Users have no idea what the data is used for, and this is the key to this law.

It's not about "cookies", that is just FUD. It's about being able to opt-in to very specific forms of gathering personal data.

Browser functionality is neither opt-in nor informed.

At least on Chrome, they are displayed precisely in the same manner as cookies.

The user doesn't have the same fine-grained control. In the case of HTTP cookies you can control whether session cookies are permitted independently of whether persistent cookies are allowed. I believe no such control exists in the domain of local storage.