Fascinating! Your camera is reporting f-numbers not only in 1/2 and 1/3 stop increments, but is also using f/5.3 and f/6, which are both about halfway between two values on the 1/3 stop scale. It is changing the value every 1/6 stop!
The only explanation I can think of is that when your D80 was serviced for aperture related issues way back in 2010, someone at the service center enabled a finer reporting scale for diagnostic purposes and then forgot to turn it back to the normal one before sending it back to you. If you ever update the firmware of your D80, you will almost certainly lose that ability.
I've long suspected that the Canon lens protocol has two aperture values that are reported to the camera: 1) The value to display and report in the EXIF 2) The more precise value a lens actually functions at when wide open at any given focal length to use for exposure calculations which are done with the lens wide open. Based on your report it seems Nikon lenses that report the aperture value electronically (i.e. "chipped" lenses) do the same thing!
There's nothing wrong with your lens. There's nothing wrong with your D3400. There's nothing really wrong with your D80, either. It just seems to more precisely report the actual aperture value the lens is telling the camera it is functioning at wide open at any particular focal length rather than the aperture number the lens is telling the camera to say it is.¹
Lensmakers fudge on numbers all of the time. But if the lenses and the cameras they're mounted on displayed the actual more precise information they use to properly function consumers would be storming the castles of the lensmakers with torches and pitchforks! If you think the difference between calling 1,000 bytes and 1,024 bytes a megabyte can cause an uproar... just try telling people their f/6.3 lens is really f/6.7 (one-sixth stop slower than f/6.3)! Or that their $11K 200-400mm f/4 lens is really a 199-389mm T4.7!!
Most 70-300mm lenses are actually 75-290mm lenses, or 78-292mm, or 73-285mm, etc. You get the idea. Even prime lenses are rounded to the nearest "standard" focal length. Who ever heard of a 292mm prime telephoto? Such a lens would be marketed as a 300mm lens. That 32mm wide angle? It's sold as a 28mm lens.
They do the same thing with maximum apertures. They almost always round the lens' maximum aperture to the next widest "standard" aperture number, which is usually a number along the one-third stop scale. When transmission is also taken into account, it's often even worse. That 50mm f/1.2? It really only lets as much light through as a theoretical 100% transmissive f/1.6 (T1.6) lens would. The 50mm f/1.4? It is really T1.7. Your 50mm f/1.8? It's actually T1.9.
¹ This brings up another thought. If we ever develop cameras to the point that their AI results in self-awareness, will they have a crisis of conscience ala HAL9000 about reporting one aperture number to the user while actually using a less desirable aperture number for internal calculations? What will be the consequences of such an occurrence? When we push the shutter button will the camera say to us, "I'm sorry, Dave. I'm afraid I can't do that?"
Note: The following information was written before the requested modifications to the question were provided by the OP. It still provides some background and a basic explanation of the difference between whole, half, and one-third stop f-number scales.
The series of full stop f-numbers, beginning at f/1, goes like this:
1, 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11, 16.0, 22, etc.
Notice that the next highest f-number past f/5.6 is f/8.0 in the full stop f-number scale.
F-numbers are based on the powers of the square root of two. The ones that are odd powers of √2 (1.4, 2.8, 5.6, etc) are actually multiples of √2 (an irrational number with a never ending decimal approximately equal to 1.414213562373095...) and are rounded to a more easily used number. For more about some of the interesting ways this works out, please see this answer to Is there a sane reason why ¹⁄₁₂₅ is not, instead, exactly half of ¹⁄₆₀?.
At one time cameras/lenses only had aperture settings (or markings for infinitely variable apertures) in full stop f-numbers. Eventually, some cameras/lenses began to allow 1/2 stop settings. The 1/2 stop f-number scale, beginning at f/1, is:
1.0, 1.2, 1.4, 1.7, 2.0, 2.4, 2.8, 3.3, 4.0, 4.8, 5.6, 6.7, 8.0, etc.
Notice that the next highest f-number past f/5.6 is f/6.7 in the 1/2 stop f-number scale.
Later on, cameras began offering 1/3 stop settings. The 1/3 stop f-number scale, beginning at f/2.8, is:
2.8, 3.2, 3.5, 4.0, 4.5, 5.0, 5.6, 6.3, 7.1, 8.0, etc.
Notice that the next highest f-number past f/5.6 is f/6.3 in the 1/3 stop f-number scale.
Your Nikon D80 allows you to select whether to use 1/2 stop or 1/3 stop increments for the f-number, shutter speed, and bracketed exposures. You can do this via Custom setting 10: EV Step. It's explained on page 89 of the Nikon Guide to Digital Photography with the D80 Digital Camera. 1/3 stop is the default and is more precise, but some older lenses with aperture rings on the lens have detents at the 1/2 stop settings. Using the 1/2 stop setting allows better metering when using such a lens in Aperture Priority exposure mode.
If the EV Step custom setting is not the issue, then the firmware in the lens is reporting f/6.7 to the camera at focal lengths longer than 360mm. Lens makers such as Tamron make the same basic design for use on various lens mount/camera systems. The same design for Canon and Nikon as well as any other mounts for which the lens is offered will have the same basic model name: Tamron 100-400mm F/4.5-6.3 Di VC USD (Model A035). The differences between one system, such as Canon, and another, such as Nikon, may require Tamron to alter the reported aperture numbers to enable their lens to better work with a specific camera maker's bodies.