Fiddler shows the filename of the most recently loaded filename in a button on the left edge of the toolbar *if* you've opened Fiddler in VIEWER mode. Otherwise the filename isn't shown by default. I'll share a script here soon you can use to add it.

Fiddler doesn't have the notion of a "currently opened SAZ" per-se, since you can open as many SAZ files as you like. What you can do is write some script that keeps track of what the most recently opened SAZ file is. Just add the code in yellow inside the OnBoot handler from Rules > Customize Rules, and add the new handler function:
    static function OnBoot() {
        FiddlerApplication.add_OnLoadSAZ(myLoadHandler);
    }

    static function myLoadHandler(o:System.Object, RSEA:Fiddler.FiddlerApplication.ReadSAZEventArgs)
    {
        FiddlerApplication.UI.Text = "Latest: " + RSEA.sFilename;
    }

// OR the C# Script version:

public static void OnBoot() {
FiddlerApplication.OnLoadSAZ += myHandler;
 }

public static void myHandler( System.Object o, Fiddler.FiddlerApplication.ReadSAZEventArgs RSEA) {
 FiddlerApplication.UI.Text = "Latest: " + RSEA.sFilename;
}

(also, note that Fiddler's LOG tab shows details of all loaded and saved files)