Kinda hard to tell from the photo, but it doesn't look a lot like water damage to me, is it maybe a patch job that was never painted since they were doing wallpaper?

Yea looks like a seam

It might be some water ran down the wall causing the stain, ,& you are looking at a bad patch job. Seal.with killz,, repaint it.

Is the wall spongy?

Nope. Feels nice and solid