I don't think this is a focus problem. Yes, depth of field is a bit shallow, the robin is more in focus than the hawk. But the odd effect is overall, occurring in both focused and OOF areas of the image.
It's not chromatic aberration. That appears right along high contrast edges of images, occurs when different wave lengths of light focus at slightly different points on the image sensor, and is either blue/yellow or magenta/green (depending upon type of CA... more info here:
https://en.wikipedia.org/wiki/Chromatic_aberration)
To me that looks like a form of "blooming", which can have several possible causes...
1. Do you have a "protection" filter on your lens? If so, remove it and do some test shots without it. A filter can cause problems like this.
2. Have you cleaned your sensor recently? In particular, oils on a sensor can cause problems similar to this. It may be lube thrown off by the shutter, for example.
3. While it's possible that VR is at fault, I've only very rarely seen image stabilization cause problems like this in images (but I use Canon gear, not Nikon). If it's caused by VR, I would suspect that the stabilization was still "working", hadn't fully completed counteracting movement yet and the image was taken while the optical elements were still moving. When I've seen something like this that I think was due to image stabilization, it always seems to be when I'm shooting really fast, trying to capture a fleeting moment before it disappears.
4. Lens out of calibration or even damaged with a de-centered element or something like that, also might cause it.
5. In this case the fast 1/2000 shutter speed should have prevented it, but I've seen similar at times caused by "camera shake blur".
6. It's also possible to produce something similar using fill flash with moving subjects, but that's not relevant here either.
I can't tell if this image is cropped or not. If there's been a heavy crop, it will make the "blooming" effect appear more severe, compared to an uncropped image. It might be happening a lot, but you might never notice it until you apply a strong crop to an image.
Regarding DSLRs focusing beyond f/8....
You need to learn how your lens works. Get out your camera. Set the lens to it's widest aperture, remove the cap and look inside the lens from the front. You might need a flash light, but will see that it's wide open. It will be perfectly round and there's no sign of the leaves of the lens aperture. They're tucked away around the sides.
Now, with the camera in either manual or aperture priority exposure mode, set a small aperture such as f/16, turn the camera around and and repeat the process of peering inside the lens from the front. It's still wide open, isn't it? Note: If your camera has it, activating Depth of Field Preview will cause the leaves of the aperture to appear and the lens to stop down. But as soon as you release the DoF Preview button, the lens will return to wide open status.
That's because all DSLRs maintain their lenses wide open until the very instant of exposure, when they momentarily stop the lens down to a smaller aperture that the user (or the auto exposure system of the camera) has set. Immediately after the exposure has been made, the aperture re-opens to its maximum.
Because of this, as far as AF is concerned, the only thing that matters is the MAX APERTURE of the lens or the effective max aperture of a lens/teleconverter combo. The SELECTED APERTURE DOESN'T MATTER (regardless whether set manually or automatically by the camera) because all DSLR lenses maintain wide open aperture until the very instant of exposure. This has been the design of SLRs since the very early 1960s, when it was sometimes called "Auto Aperture" (no relation to "auto exposure"). Prior to that, and even with some lenses afterward, the user had to manually stop the lens down just before the exposure and it didn't reopen automatically. In fact, back then during the Jurassic Period, when we were photographing dinosaurs on something called "film", with many cameras we also had to manually cock the shutter to make the mirror finish it's cycle and drop back down. The mirror remained up, blocking the viewfinder until the camera was reset by the user. Hey! We've come a long way, baby!
Most DSLRs' viewfinder-based, phase detection AF systems are "f/8 limited"... quite a few are even "f/5.6 limited". This means that in order for AF to have enough light to work the max aperture of the lens must be at least f/8 or f/5.6, respectively. Some are limited to only using one particular AF point, or a smaller than usual selection of them, when an f/8 lens or lens/TC combo is being used.
The Nikon 200-500mm is an f/5.6 lens. If a 1.4X teleconverter is added to it, the combo becomes an effective "280-700mm f/8". That's still able to autofocus on some cameras, though it may be limited to only the center AF point. But if a 2X teleconverter were instead used on that lens, the combo would be an effective 400-1000mm f/11... which virtually no DSLR can autofocus.
It's also important to realize that each full stop reduces the amount of light passing through the lens by one half:
- f/1.0 passes 100% of light to pass through.
- f/1.4 passes 50% of light through.
- f/2.0 passes 25% of light.
- f/2.8 passes 12.5% of light.
- f/4.0 passes 6.25% of light.
- f/5.6 passes 3.13%.
- f/8.0 passes 1.56%.
- f/11 passes 0.78%.
- f/16 passes 0.39%.
and f/22 only allows about 2 tenths of one percent of light to pass.
Note: There are some lenses that "push" the limits of AF with an f/6.3 aperture. Those lenses are designed to "fool" cameras that are "f/5.6 limited" into still trying to focus. Auto focus is likely to slow and/or "hunt" more, whenever there's less light to work with. It will vary, too, depending upon ambient lighting conditions, detail and contrast of the subject, and more.
Also, some mirrorless cameras now have ability to focus much lower light conditions and "slower" lens/teleconverter combos. For example, the Canon EOS R can focus effective f/11 or even f/16 combos (Ken Rockwell proved this with tests using both 2X and 1.4X teleconverters on an f/5.6 lens.) This is possible because most mirrorless use a different type of auto focus system that's embedded in the imaging sensor itself. It may be possible to use certain DSLRs Live View mode similarly.
I don't think this is a focus problem. Yes, depth ... (