Thanks for reaching out. While Zen Browser is not a supported browser you should be able to get app integration working using similar steps to the ones that you posted. Can you try the steps again in this order:
Ensure that the browser integration is enabled from Settings → Browser within 1Password for Linux.
Close 1Password completely.
Pop open a terminal, and run sudo mkdir /etc/1password.
Run cd /etc/1password.
Run sudo gedit custom_allowed_browsers.
Paste in the appropriate browser code - such as opera or vivaldi-bin. (the binary name)
Save the file.
In terminal, run sudo chown root:root /etc/1password/custom_allowed_browsers && sudo chmod 755 /etc/1password/custom_allowed_browsers
Run 1Password - it will read our new config file and make the appropriate connections.
Launch your browser.
Make sure that you place the actual binary name into custom_allowed_browsers and not just a symlink pointing to the binary.