I don't remember exactly the sequence of events, but I tried to update stock apps like Notes, Numbers, and Garage Band first before updating to iOS 10.
Now I am using iOS 10 smoothly but just noticed that Notes does have share notes (collaborative editing) feature added with this release. I went back to the App Store to find only an "Open" option for notes, and not update or download.
I noticed though I have Numbers and Garage Band on my phone, but it showed in the App Store that they need to be downloaded again.
I am new to iOS.
Best Answer
Issue was not with the Notes app but the way I was using it. Collaborative edit is only possible in notes on iCloud and not "On My iPhone" ones. I had all my notes on the latter, so was not seeing the latest and greatest features.