After another hour spent today wondering, I finally realized to try another USB cable. I think all my issues were coming from a defective cable.
With the new cable and the Gpsmap 65 in MTB mode I get the following results with sudo dmesg -w which @peregrino69 suggested.
$ sudo dmesg -w | grep Garmin
[366555.328607] usb 1-1: Manufacturer: Garmin
[366564.028948] scsi 1:0:0:0: Direct-Access Garmin Garmin Flash 1.00 PQ: 0 ANSI: 5
[366564.029569] scsi 1:0:0:1: Direct-Access Garmin Garmin SD Card 1.00 PQ: 0 ANSI: 5
The device now also shows up with
$ lsblk
NAME MAJ:MIN RM SIZE RO TYPE MOUNTPOINTS
sdb 8:16 1 14,6G 0 disk /media/username/GARMIN
Now I can also copy the tracks from the device. They are in the Garmin/GPX directory. Here is the directory structure just for reference:
/media/username/GARMIN
├── Garmin
│ ├── BirdsEye
│ ├── CONNECT
│ ├── CustomMaps
│ ├── CustomSymbols
│ ├── Debugging
│ ├── EDM
│ ├── Express
│ ├── ExtData
│ ├── Filters
│ │ └── Geocache
│ ├── GPS
│ ├── GPX <-- Here be tracks ;)
│ │ ├── Archive
│ │ └── Current
│ ├── Profiles
│ ├── SQL
│ │ └── temp
│ └── Text
└── System Volume Information