Unique form per user with custom form class
Your updated post shows the issue. The issue here isn't the form, it's the underlying storage method. You are using configuration, which is for site configuration, ie configure how the site as a whole works. But you want to save values per user, not for the whole site.

What you are looking for is having a form used by all users, saving the submitted values for the user, rather than for the site.

I would suggest using the user.data service. This service lets you save data for users. You can user UserData::set() to set values, and UserData::get() to get values. Get the values in the form definition to get the default value, and set the values in the submit handlers to update them with the current values.

As a side note, if you were creating a form for configuration (which you're not), you would want to extend ConfigFormBase, which it doesn't appear you were doing, based on your code.