Airdrop does not recognize my iMac

Boot into Safe Mode by booting with the Shift key held down and check there to see if the problem persists.  Reboot normally and test again.

NOTE: Safe Mode boot can take up to 10 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

Aside from all the other suggestions, my personal experience is: the newer the device, the better the chances of Air drop actually working. I had a 2012 MBP and could not get that to be recognized most of the time - it would only work about 10% of the time. Your Mac is a 2013. My 2017 iMac is recognized about 75 - 80% of the time, but is temperamental enough to not work 20 - 25% of the time.

In other words: in my experience, it is not 100% reliable.

Try rebooting your router

What year/model is your Mac?

And OS version?

And, what year/model is the iDevice you are trying to use?

Here is the Apple Support document on using your 2013 iMac with your iPhone and iPad.

https://support.apple.com/en-us/HT203106