Thanks for your suggestion but there appears to be something more software-oriented in play here with respect to the 2nd gen nano, and in fact specifically with reference to that generation of nanos, since iTunes 12.9.x still "plays nice" with some other ones, e.g., 6th and 7th gen.
A little while ago today I thought to try the 2nd gen nano in question with a spare machine, my mid-2010 MacBook running El Capitan 10.11.6 and iTunes 12.8.2.3.
I have the 2nd gen nano's preferences set to open iTunes when I connect and also set to manage music manually. So I booted the 2010 MacBook and when I connected the 2nd gen nano, it was recognized immediately, triggered a launch of iTunes and mounted on the desktop with its nickname.
All that was what I would have expected it to do also on the MBP running Mojave and iTunes 12.9.x... But, the 2nd gen nano still won't connect to the MBP or at least is not recognized as doing such when I look at info from the USB tree in the system report for that machine.
Since I have not updated these nanos in a long time it's not clear to me from a successful connection to the 2010 MacBook this morning whether the problem on the MBP is related to iTunes 12.9 or to the Mojave OS on that newer machine.
Ugh... I guess I could sneaker-net a selection of music from the MBP over to the MacBook and then update the nano --and its companions, since I was thinking to refresh the contents of three or four of these 2nd gen nanos with different music when I first set out on this mission that turned into "a project" yesterday afternoon.
I don't regard this problem as solved. Not least because the 2nd gen nanos are still going strong, the MacBook is an older machine and my MBP although a mid-2012 model, was bought refurbed directly from Apple in 2016 and is still under AppleCare warranty so I'd expect that MBP model to remain at least my secondary machine into the 2020s since it's built like a tank.
All that said I realize the ipod nanos have been discontinued by Apple and carry assorted designations (supported, vintage, obsolete) which appear in practice not to be correlated in any particular way to how they work with iTunes.
Still I am mystified by any intentional removal of iTunes compatibility to something as relatively simple as a 2nd gen nano. Those little devices are practically indestructible too, so it's not like Apple should be surprised there are so many of them still in use!
And... as I said in my earlier post, my iPod Classic still connects to iTunes 12.9.x -- and so does my 6th gen nano, and my 7th gen nano. Both of the latter nanos get updated fairly often. These devices have more complex features than does the 2nd gen nano, so the disdain of iTunes 12.9 for the 2nd gen nano is quite puzzling.