To permanently allow a user to install updates via update manager in Linux Mint, a policy needs to be added to PolicyKit (polkit):

sudo tee /etc/polkit-1/rules.d/99-mintupdate.rules > /dev/null << 'EOF'
polkit.addRule(function(action, subject) {
    if (action.id == "org.aptkit.install-or-remove-packages" &&
        subject.user == "username") {
        return polkit.Result.YES;
    }
});
EOF

then either reboot or restart the polkit service:

sudo systemctl restart polkit