Yeah, same issue, terrible UX, I don't understand how no fullscreen playback option, for iPad, made it to final release. This will look really bad on the iPad Pro. 😉
No fullscreen video podcast play back, giant bright white bars taking up half the screen is just bad U.I. design.
Having a light or white interface in a video play back app is wrong and then not having a proper full screen option is just a major fail. Makes no sense to not have it and I really don't understand the bright white U.I. background. When I go to a movie theater, the room is dark, not lit with bright white lights, during the movie.
I also don't like the translucent white U.I. background of the Video app either, but at least it fades away as your video plays in fullscreen.
I really hope this gets fixed because it should be considered a bug. I did a quick web search and found a few other mentions of this issue, some people reported it the bug during the iOS 9 beta and one person noticed the issue still exists in the current iOS 9.1 developer beta.
http://forums.imore.com/ios-9/338585-ios-9-beta-podcast-ipad-no-full-screen-vide o.html
Send Apple feedback so they fix this Podcast app bug
http://www.apple.com/feedback/podcasts_ios.html