THIS BUG IS FIXED IN 21w10a
The fog is not rendering in a smooth circle on AMD cards and integrated graphics even tough is supposed to be solved (the fancy option isn't available).
This feature is working correctly only on Nvidia graphics cards.
More info:
1. MC-32452
Update: Sodium, a mod made by jellysquid_ fixes this issue. Fog is rendered properly when using this mod on AMD graphics cards and integrated graphics.
(Link to the mod: https://www.curseforge.com/minecraft/mc-mods/sodium)
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OpenGL: Intel(R) HD Graphics 4600 GL version 4.3.0 - Build 10.18.14.4170, Intel
Outdated Intel drivers, update to V15.xxx: https://downloadcenter.intel.com/download/25489/Intel-Graphics-Driver-for-Windows-10-15-40-4th-Gen-
OpenGL: GeForce GTX 860M/PCIe/SSE2 GL version 4.5.0 NVIDIA 358.91, NVIDIA Corporation
This is one version behind (new drivers released two days ago)
Please update your cards to current drivers and retry.
Updated both to latest - no change visually. The only difference I can see between the cards is Intel's GL version 4.3.0 vs NVIDIA's GL version 4.5.0 but I have no idea what difference that would make if any.

Ok, it was worth a try.
Can confirm this is happening in 1.10.2. Fancy fog uses a function only available on NVIDIA drivers.
Affects:
1.11
1.11.1
1.11.2
1.12
1.12.1
Using optifine on linux( fedora 26 with 4.12 kernel, mesa 17.1.5) on an amd dedicated card( R9 m265x, Cape Verde) and an intel integrated card( hd 4600), I can see fancy fog being denser on the amd card vs the intel card( the intel one doesn't seem to do anything different on fast vs fancy) but not as dense as the screenshots on OP. Same goes for the generic fog on vanilla but I can't find the setting to change that anywhere on the video settigs. Here are the openGL Core/ES/Extension versions supported by both cards:
[headsuphigh@192 ~]$ glxinfo | grep "version"
server glx version string: 1.4
client glx version string: 1.4
GLX version: 1.4
Max core profile version: 4.5
Max compat profile version: 3.0
Max GLES1 profile version: 1.1
Max GLES[23] profile version: 3.1
OpenGL core profile version string: 4.5 (Core Profile) Mesa 17.1.5
OpenGL core profile shading language version string: 4.50
OpenGL version string: 3.0 Mesa 17.1.5
OpenGL shading language version string: 1.30
OpenGL ES profile version string: OpenGL ES 3.1 Mesa 17.1.5
OpenGL ES profile shading language version string: OpenGL ES GLSL ES 3.10
You can also find every extension supported per driver on www.mesamatrix.net
Using optifine
Optifine changes the way Minecraft graphics work. It is irrelevant to this ticket as this is a bug tracker for unmodified Minecraft only.
As I mentioned the issue is present on vanilla when comparing the intel and the amd card. Optifine suffers from it too but less. Is there any way in vanilla 1.12.1 to change the fog density so I can do more comparisons ?
Affects 1.12.2-pre1, 1.12.2-pre2, 1.12.2, 17w43a, 17w43b.
Affect 1.13-pre2
Affects 1.13 and 1.13.1.
In 1999 Nvidia created a standard known as "nVidia Distance Fog" which is only compatible with their cards. When using a Nvidia GPU this gets enabled by default but isn't supported on AMD and Intel GPUs. You can disable it in OptiFine by setting "Fog" to Fast.

Yes, that's what I am saying it's working as intended on Nvidia Graphics cards.

fixed in 21w10a

Yes, it is in fact fixed! Really glad to see this bug fixed!
For both cards:
Please force a crash by pressing F3 + C for 10 seconds while in-game and attach the crash report (
[minecraft/crash-reports/crash-<DATE>-client.txt|http://hopper.minecraft.net/help/finding-minecraft-data-folder]
) to this ticket.