If you have access to a browser, you have no protection.
Anyone connecting to your site from a custom browser can simply type console.log(localStorage) in the developer tools (use Ctrl-uppercase-i in most browsers) to see it explicitly.
The solution may be to encrypt data using the key provided by the server, but it will not be so secure: it is easy (for example, using the extension) to change the javascript executable as soon as you get access to the browser (and you ask the user to return to the site). I would suggest storing this data on the server.
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