It appears that you cannot share notes via Airdrop from a Mac.
I have checked four Macs (all running either El Capitan or Sierra) and in each case I find the exact same situation as you. More specifically, trying to share a note does not give me the option of using Airdrop and if I click on the More... option I also see AirDrop listed with a ticked checkmark (greyed out just like in your screenshot).
This is not something I had noticed previously as I haven't tried to share a Note from any of my Macs.
Now, I also checked whether this is a known issue and found this discussion on Apple's website. Although it was started in January 2015, you'll see that the most recent post was (at the time of writing this) just over a week ago.
I'm not entirely sure why sharing notes via AirDrop from a Mac isn't an option (although the fact that notes are all stored within the Notes app and can be synced via iCloud may be part of the reason).
I suggest you send feedback to Apple.
If you do need a way to share your notes via Airdrop, then there's a few workarounds online (including within the above-mentioned discussion). The one I like most is to do as follows:
- Within the Notes app, open the note you want to share
- Go to File > Export as PDF...
- Click Save
- Within Finder, navigate to the exported PDF
- Right-click on the file
- Select Share > AirDrop