Native feel is not wanted (anymore). Branded appearance is.
(and I think this is unfortunate. I wish developers didn't attempt to replace my browser's perfectly fine-looking, accessible checkboxes by their own often ugly custom implementations for instance. And I'm not hating on web devs, I'm one of them)
XUL [1] seemed like a good idea as a basis for building native-looking web applications. It's too bad it has been abandoned.
I think both are wanted. People like it when they have the usability and familiarity of native widgets, but it’s nice when they also have theming and personality. A good example is TweetBot, which is a normal iOS app except it’s not, or OmniFocus which does things in a native way except everything is purple.