the law covers storing data on a user's computer. If you record their personally identifying information (i.e. user agent) on your computers, then that might fall under the decades old data protection directive in EU.
Also if someone does not consent to storing data on their computer and it's necessary to store it to get it to work, then they cannot use your site. The law doesn't require that your site work fine without local data. That would be silly.
Also if someone does not consent to storing data on their computer and it's necessary to store it to get it to work, then they cannot use your site. The law doesn't require that your site work fine without local data. That would be silly.