Is it possible to change NavigationView's settings item's icon and text?

Yes, you can edit the default template of the control to modify the style of the settings area. You can find the following there:

<NavigationViewItem x:Name="SettingsNavPaneItem" Grid.Row="5">
    <NavigationViewItem.Icon>
        <SymbolIcon Symbol="Setting"/>
    </NavigationViewItem.Icon>
</NavigationViewItem>

So changing the icon is easy. Regarding the text you may be able to change it here as well, but I am not sure if the control will not override it and can't verify as I'm not at my PC. Try to use the SettingsItem property of the NavigationView and cast to NavigationViewItem.

Update: Changing the text

Here is how you can customize the settings item text, I don't know why I haven't thought about it before - create a custom NavigationView and update the text manually in OnApplyTemplate:

public class CustomizableNavigationView : NavigationView
{
    protected override void OnApplyTemplate()
    {
        base.OnApplyTemplate();
        var settingsItem = (NavigationViewItem)GetTemplateChild("SettingsNavPaneItem");
        settingsItem.Content = "Custom text";
    }
}

Update: WinUI

Going forward, the recommended solution will be to use the WinUI library and its NavigationView control. This allows you to add multiple NavigatoinItem in the PaneFooter area and those are fully configurable. In addition to this the new control supports not only classic hamburger menu display but also can display horizontally on top of the view, which is now the recommended navigation pattern on large screens.