Cannot upgrade iPad Air 2 from iOS 9 to iOS 13
I'm not sure why iTunes recommends iPadOS (iOS 13 equivalent for iPad), but it won't update as the iPad 2 is not compatible with iPad OS, and the highest software it can go to is iOS 9.3.5.

Here is a list of compatible devices on the Apple website:


  
  12.9-inch iPad Pro.
  
  
  11-inch iPad Pro.
  10.5-inch iPad Pro.
  9.7-inch iPad Pro.
  iPad (7th generation)
  iPad (6th generation)
  iPad (5th generation)
  iPad mini (5th generation)
  iPad mini 4
  iPad Air (3rd generation)
  iPad Air 2
  
  


You should be able to update to iOS 9.3.5 using the standard Software Update page (accessed by going to Settings -> General) over the internet without a computer.

The only thing you'll get from updating from iOS 9.3.1 is some security fixes, so leaving your device at that isn't a major issue.

Solution for OP

The solution was to restart iTunes on the Mac.