OK - one solution laid out a in a manner more easily understood…
The icons that each program ‘owns’, and is displayed on such things as docks, are directed by a file called application_name.desktop. What we are going to do here is move a copy of that file to your local directory, where its setting will have priority over the system settings when you are logged ion. First we make sure that the directory ‘applications’ exists in ~/.local/share. Create it if it’s not there. Then
cp /usr/share/applications/chromium-browser.desktop ~/.local/share/applications
Check that it made it there. Now it needs some editing. You could use the sed command given in that post, or just open the file with a text editor, and add the line in yourself. Depending on your DE, you may be able to do it by editing the Properties tab in your file browser instead. (edit: I see you’re on XFCE, so that is a good choice to use) Any which way, add this:
--password-store=basic
either in the box for command in Properties/Lancher, or between the Exec=chrome-browser and the %U, with spaces surrounding it.
I am not sure of the exact contents of your Exec= line, as I don’t run Chrome, but it should be clear where to put it if you’re hand-editing.
Once this is done, the next time you start it up, there should be no demand for a password - but also no extra protection for the passwords stored in your browser. Which may not matter, if your main user password is good, of course!
Hope this helps, and clarifies what is suggested in the past.