I also have an Edge Touring. A couple of comments;
As this is an obsolete model (by Garmin's standards, not mine), I'm surprised that there is any new software for it, apart from map updates.
Although I don't use Linux, I find that Garmin devices are very fussy about which USB port and which cable you use. Sometimes they show up as devices and sometimes they don't.
And yes, it does, or should show up as a mass storage device (on Mac and pc at least)