Digital Editions does not generally recognize tablets with eReader capabilities.
You can transfer books to the mounted device using a file manager such as Windows Explorer or Mac Finder.
As long as you authorize the device with the same Adobe ID as you use for ADE,
you should be able to read the books fine.
I'm not 100% sure if Kobo's builtin eReader app supports Adobe DRM; if not try installing the free Bluefire or Aldiko app for reading.