When I re-visit a protected category, I'm not asked to re-enter a password
When you enter the correct password for a password protected category, then a cookie is set in your browser. This unlocks the category for the number of days specified in the WooCommerce Protected Categories plugin settings. This means that if you revisit the category from the same web browser then you will not be asked to enter the password again.
This is intended behavior and saves your customers having to re-enter the password within a short period.