I was on some site with wallpapers. And, there is download button there below specific image.
But OK, if it's a feature that was introduced in the latest version, what's the point then having that option in opera settings?

If it's feature then is really retarded feature. And, I don't think it's a feature, I think it's a bug and they need to fix it. Everything worked fine on previous versions of Opera. I noticed yesterday same thing as you, and I keep asking myself what's wrong...
I tried to download .jpg file and some other files - result was the same. Opera keeps asking me where I want to download my file. With version 37 I had no issues with that. So, definitely it's a bug and they need to fix it soon cuz it's so annoying.