First we start by creating a wineprefix and installing our prerequisites from terminal:
WINEARCH=win32 WINEPREFIX=/home/$USER/GarminExpress winetricks dotnet452 vcrun2010 corefonts
WINEARCH=win32 WINEPREFIX=/home/$USER/GarminExpress winetricks win7
Note: You will be prompted during the setup for .NET installation just hit next and finish. Also, I like to seperate win7 after the prerequisites are setup to make sure the wineprefix runs in Windows 7 mode.
With that out of the way we need to download and install our GarminExpress.exe Setup executable. I downloaded mine from https://www.garmin.com/en-US/software/express
Run the GarminExpress.exe file in our wine prefix
WINEPREFIX=/home/$USER/GarminExpress wine explorer /desktop=garmin,1366x768 /home/$USER/Downloads/GarminExpress.exe
Note: If you downloaded the GarminExpress.exe to another location replace it above
Modify your shortcut to be the following command
WINEPREFIX=/home/$USER/GarminExpress wine explorer /desktop=garmin,1366x768 ‘/home/$USER/GarminExpress/drive_c/Program Files/Garmin/Express/express.exe’
Upon Launch you should see this screen which should recognize your Garmin device
Was able to finally get a Windows 8 VM up and running for comparison, and maps do update, so it does appear to be a Wine issue (or maybe a Linux USB issue judging from some stuff I've read online?) -- not a device issue. That being said, Garmin Express also appears to be just kind of wildly temperamental in general -- it refused to install until I updated Windows, then refused to boot after I installed it until I uninstalled it, cleared out its cache files, and then reinstalled it... so maybe it just doesn't like my computer/distro.
The upshot is that if you're finding this thread, MJWootton's comment works to install as of October 2022, and you should be able to at least download new firmware, but you may not be able to update maps. Maybe you'll have more luck though because this software appears to just randomly break sometimes.
Given that Garmin GPSes just mount as USB drives, it's really ridiculous that you need software to update their maps in the first place and that there isn't a manual process for doing so. But companies are going to do their thing I guess.