Hmm.. I am not sure to what degree a "File Open" dialog is either Caja or Nautilus. Honestly, I have not used Nautilus in a while so, I'm not sure what it looks like but, I think you are describing the difference between GTK2 and GTK3.
You can see some of those differences posted here:
The applications you specified "Google Chrome, Firefox or Opera" are just compiled for GTK3 and not GTK2. In Ubuntu MATE 16.04 LTS, I think that most applications included in the OS are still compiled for GTK2. Moving forward in MATE, and someone correct me if I am wrong, GTK2 is being replaced by GTK3. So, most file dialogs, if not all, will appear as they do in the screenshot you posted.