Hard to tell, but, I think maybe better to learn to love the texture, and just repaint it. Looks like it could be a big mess, if you try to take it all off. It doesnt look quite right to me. I wouldnt take anymore off the painted surface. I see you have at least one spot there, and to get the same texture back on that spot if you have gotten below it, will not be easy. If that is raw plaster, it is rough, and you couldnt paint it either, without having it skim coated. Its not a bad texture. Its fairly subtle.

Thanks Cat...the white is definitely where the drywall surface paper has peeled off. At the very least it needs to be skim coated for sure. We wanted to find out if it's wallpaper because we were planning to do some tiling on that wall, and it seems like a pretty bad idea to put tile over wallpaper! FYI...it's a powder room, and this tile would be in a dry spot, so no worries on that account. Just decorative.

I suspect, whoever wallpapered it the first time, put the wallpaper directly on drywall, and thats why the drywall paper came off. Its a huge job, to get wallpaper off, when someone hasnt primed the walls before putting wallpaper up. No, you shouldnt tile over wallpaper It would have to be removed. My big question is though, do you actually have drywall, or is your house plastered? That picture, doesnt look like drywall, where its white.

To me, it looks like wallpaper over dried drywall mud with the drywall behind the mud. Possibly the towel bar or whatever was there was replaced at some time in the past and the old holes were patched?