I see from the tutorials that you can round the edges of the rectangular picture-in-picture border.  But, is there a way to have the picture-in-picture actually appear in a round or oval border?   See attached photo for example of what I mean.

Have you tried MultiShape PIP?

Click on asset on the timeline. Effect>2D-3D>MultiShape PiP.

In "Select preset">Soft Oval PiP". Then customize. You will have to move the sliders to get it in position along with everything else you have to adjust. But it works.