(This answer is based on the assumption that you are not using different "protective" UV filters, ND filters, Polarizing filters, or any other type of filters on either lens. If you have different filters on each lens, it should be rather obvious where the differences are mostly coming from.)
Why is a lens darker than other ones when applying the same settings?
The most likely explanation is that the 18-105mm lens with mechanical aperture control is incorrectly exposing lighter than the 16-80mm lens with electronic aperture control.
The difference is subtle, but significant.
That is to say, the electronically controlled aperture of the 16-80mm lens is probably giving you more accurate exposure than the mechanically controlled aperture of the 18-105mm lens.
If this is happening with all of your DX lenses, then the issue is most probably in the camera's mechanical aperture linkage, rather than in the linkages of the DX lenses. If it is also happening with other camera bodies, then chalk it up to the general differences between mechanical aperture control and electronic aperture control. Or maybe the linkage on your friend's D3200 is as worn or has been bent by about the same amount as your D500.
A little background¹
When AF technology began emerging in the late 1980s, Nikon attempted to create a system that would allow old F mount lenses all the way back to the late 1950s to remain usable as manually focused lenses on the new AF capable bodies. They chose to place the focus motor in the camera where it drove the focus elements in the lens via a mechanical linkage, rather than place the focus motor in the lens. In addition, they chose to retain the mechanical linkage between the camera and lens to control the aperture and associated metering so that it would be backwards compatible with older F-mount lenses. Pentax took this approach as well.
A couple of other major camera manufacturers chose to make a clean break and create a new lens mount system with an all electronic connection between the camera and lens and to place the focus motor in the lens. Minolta introduced a new 'A-mount' with an all electronic system in 1985 (this eventually became the Sony A-mount after Sony bought Minolta). Canon introduced the similar EOS system in 1987. Neither system allowed users to use previous lenses in older mounts bought from Minolta or Canon, respectively, with the new cameras that used the new mounts. Early on, Nikon gained market share by making their new AF cameras and lenses backwards compatible with existing F-mount cameras and lenses.¹
For most of the period since Minolta (1985) and Canon (1987) introduced camera systems with an all electronic mount, Pentax and Nikon have gradually introduced electronic connections to their existing mount systems in several piecemeal stages. Pentax did so sooner and more aggressively than Nikon.
Soon, the new "Ultra-Sonic Motor" design Canon used on all but their low end lenses proved to be far superior in terms of autofocus speed and accuracy when compared to the mechanical linkage that Nikon, Pentax, and others used. Almost overnight Canon captured much of the professional 35mm market that Nikon had dominated for decades, particularly among those who shot sports/action. In order to remain competitive, in the middle 1990s Nikon added electrical contacts to their F-mount system and began creating AF-I lenses with motors inside them for large telephoto lenses that require heavier focus elements. AF-S lenses with AF motors that were designed very similarly to Canon's ring type USM didn't appear until 1998. Nikon continued to place AF motors in their bodies as well to drive the existing AF lenses that lacked their own motor. (Only with the introduction of entry level digital SLRs did Nikon introduce AF era F-mount bodies that did not have AF motors in the camera. The D3xxx and D5xxx bodies can only AF with AF-S lenses or the even newer AF-P lenses.)
But Nikon continued to offer only mechanically controlled apertures in all of their lenses until well into the 21st century.
Other than a few Perspective Control (tilt/shift) lenses introduced in 2008, Nikon did not offer an F-mount lens with an electronically controlled aperture until the AF-S 800mm f/5.6E VR in 2012. Several other high end (and expensive) 'E' lenses followed.
The AF-S 16-80mm f/2.8-4E Dx VR was the first 'E' lens from Nikon that did not cost upwards of around $2,000. It was rolled out in the second half of 2016, around thirty years after the first mass-consumer lenses with electronically controlled apertures. In the intervening years several other new mounts/systems had also been introduced that use only electronic, rather than mechanical, communication between the camera and lens. Among them: the Four Thirds and Micro FourThirds system from a consortium formed by Olympus and Panasonic, Sony's E-mount, Fuji's X-mount, Samsung's NX mount (now defunct), and even the compact Nikon 1/CX mount (also now defunct) announced in 2011.
As cameras that utilize all electronic camera/lens communication began being used for purposed not even dreamed of in the mid-1980s, the advantages of electronically controlled apertures became more and more apparent over the three decades between the mid-1980s and the mid-2010s:
- Faster actuation. The servos used in electronic lenses are more compact and there is significantly less total slack in the system. With no return springs, the servos can also open the aperture after the exposure as fast as it was stopped down.
- Less susceptibility from very cold temperatures slowing stop-down immediately before an image is captured.
- Better shot-to-shot accuracy when both systems are new and properly adjusted.
- No need to periodically test and adjust the linkage mechanisms on both the camera and each lens as they wear and/or as adjustment screws loosen.
- Lack of susceptibility to the mechanical linkage being bent when the lens is attached to the camera. If the camera's lever is bent, it will be inaccurate with all mechanically controlled lenses used with the camera. This usually manifests itself with overexposure.
There's also the possibility that 35mm, which seems to be the sweet spot for the 18-105mm lens' f-stop to T-stop ratio when wide open, is also a focal length where the 16-80mm lens may have a larger difference between f-number and T-stop. Even though you are using both lenses at f/8, most lenses tend to "preserve" differences between the specified f-number and the actual amount of light transmitted by a lens as it is stopped down. Lensmakers do this to maintain the distance between each stop in the range of aperture settings. With zoom lenses, it's more common to see differences between f-number and T-stop when the lens is wide open and the focal length is changed.
Here's the transmission profile for the AF-S DX 18-105mm f/3.5-5.6 G ED VR (Orange) and two other Nikon lenses published by DxO Mark (unfortunately, neither DxO nor Imaging Resources have published measurements for the AF-S 16-80mm f/28-4E ED VR):
What we would expect in the upper chart for a "theoretical" 18-105mm f/3.5-5.6 is a line with a more or less constant slope from somewhere slightly darker than T-3.5 on the left to about the same amount of slightly darker than T-5.6 on the right. That's what we do see with the AF-S 24-120mm f/3.5-5.6G IF-ED VR (Blue). There's very little difference between rated f-number and measured T-stop across the entire zoom range for the 24-120mm f/3.5-5.6. But that's not what we get with the 18-105mm.
Note that a few other Nikon DX zoom lenses, such as the AF-S 18-135mm f/3.5-5.6G IF ED (not shown) and the AF-S DX 18-70mm f/3.5-4.5G IF ED (Red) have an almost identical profile compared to the 18-105mm. It seems that with some of the lower cost DX lenses, Nikon is closing the wide open aperture down just a bit at the wider angle focal lengths, perhaps to limit aberrations on the edge of the image field?
Without T-stop measurements for the AF-S DX 16-80mm f/2.8-4E ED VR, it's hard to say if the difference you are experiencing can be attributable to that lens having a higher T-stop value when zoomed to 35mm. It might be interesting to try a similar test using 16-18mm, 50mm, and 70-80mm with each lens to see if the results are the same as at 35mm.
¹ For an even more extensive look at the history of the Nikon F-mount and how it compared to competitors' mounts since the introduction of AF in the 1980s, please see this answer to another question.
² The digital revolution made small increments of exposure variation more of an issue than with film. As time lapse photography and video using cameras primarily designed to make still images became more common, this proved more and more significant.