It is a feature of the Kindle reader software, not the ebook.
Which begs the question, if you are asking about a feature of Kindle software, why are you posting in the EPUB format subforum?
Either way, the "magical" solution (magical solutions rarely exist) is to develop an alternative firmware for the Kindle, or at least a portion thereof. You could actually do it via a custom ACX, dsmid has written documentation on how to develop custom ACX widgets...
Of course, it also requires every fan to jailbreak their Kindle.
And running a server to collect that information.
The Facebook sharing at least is easier, there is actually an option in calibre's MOBI/AZW3 conversion dialog to enable that. Again, it isn't part of the book, it is part of the Kindle... but it is activated for books which have a mobi-asin identifier like Amazon purchases have.
...
Are you beginning to see how not-easy this is?
It would make a lot more sense to do something perverted with EPUB3 (which will only work on ereaders which support EPUB3, which means you are targeting tablets, or maybe the Kobo) involving a lot of javascript... to which I can only say that ebooks are not meant to contain the ereader logic and/or phone home. (Can you tell I disapprove of EPUB3?
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At a certain point you have to ask yourself whether it makes more sense to implement error reporting inside your own web novel reading app.
