For Mobipocket format books, the Kindle actually stores your notes in a separate file with an .mbp extension for each book, so you can move your notes with the actual corresponding book file between readers.
However, they've changed the .mbp format so that it no longer stores the text of the highlights and have done something so that files created on a hardware Kindle can't be read on a Kindle for App unless you set them up in a particular manner. See this thread
in the Mobi/PRC format forum for details.
The actual hardware Kindle does auto-export your highlights and notes the first time you make them to a text file on the Kindle, and Amazon does auto-backup for notes associated with your paid purchases onto their website. But you can't re-export your highlights/notes on your non-Amazon books at will, as far as I can tell.
If you can still pull data off the drive of your Sony/Kobo, it may be worth your time to ask around in those respective forums if anyone knows where the highlights/notes are stored. IIRC Sony keeps them in a particular file which you may be able to try and retrieve and see if it's exportable in some fashion.
Hope this helps.
ETA: Apparently Sony keeps the full text of your notes and highlights in an SQL database which you can manage with standard tools? This thread
in the Sony forum may be of help.