How to edit shared iCloud note in native app?
Solution: Upgrade to a macOS version which introduced the support to do so. According to the About collaboration for Pages, Numbers, and Keynote article on the Apple Support pages "A Mac with macOS Sierra and Pages 6.3.1, Numbers 4.3.1, or Keynote 7.3.1 or later" is required.

In my case the macOS upgrade from El Capitan to High Sierra provided the desired feature. I simply did not think about the only recently introduced software support.