Unlike the paper book, the ebook has some advantages (searches, for example) but it also has limitations. The impossibility of dedicating it (by the author or the person who gives it) is one of these.
Would the Mobi format support a "signing book widget" (a blank page that allows writing by hand) to supply this limitation?
No. There used to be a signing widget, but I am pretty sure it predated Kindle Direct Publishing (formerly Digital Text Platform, founded November 2007). And the signature was phony -- it wasn't mine, as I recall.
(Don't trust KDP to publish a print edition. Don't trust CreateSpace to publish an ebook. Each does one thing well and the other thing poorly.)