file-picker-dialog
kate uses kde5 file-picker-dialog
You should install oxygen icon pack:
sudo apt-get install oxygen-icon-theme*
and if the left side panel (file list..) is missing:
sudo apt-get install kdelibs-bin kdelibs5-data kdelibs5-plugins
"Kate uses QT file dialog not KDE's"
The only workaround I've found is to use Konqueror to browse the filesystem, and then use the right-click context menu to open the file with Kate.
Note that installing Kubuntu-desktop and/or Plasma instead of Unity does NOT fix the problem with Kate.