Does Skyrim take advantage of 2GB video cards?

Yes, the game will use more VRAM. I don't know if there's a limit; I'm unaware of anyone using, say, a Titan RTX with its amazing 24 GB of VRAM and then trying to load everything with 4K and 8K textures to see if they can fill it all. However, the original game (Skyrim Legendary Edition AKA "Oldrim") can only use about 3GB of system RAM, because it's a 32-bit application. So more system RAM is better, but only up to a point, after which it has no effect. You need as much RAM as will give the game at least 3 GB, after the OS and whatever other processes have gotten theirs. And you need as much VRAM as you can give it, because when all VRAM + 3 GB RAM both fill up, the game then crashes to desktop. If you use a lot of mods or improved textures, you'll definitely feel the pain with a GPU that only has 1–2 GB VRAM. Skyrim Special Edition (SSE) is a 64-bit app, and doesn't have that RAM limitation, but is still VRAM-hungry.

Next, you should actually cap the FPS slightly lower than 60, like 57–58 FPS, because if you set it at 60, it can still spike over 60, and the game's Havok physics engine can still flip out. People who claim "I run Oldrim at 120 FPS and it's just fine" either haven't been running it long, or are having the physics flip out but in ways they don't realize the game is not supposed to be behaving. The earlier game, Oblivion, had similar issues (the one before that, Morrowind, did not, because it used a different physics engine). I gather that SSE doesn't have this framerate-and-physics limitation issue, but I have not verified this with direct testing.

In Oldrim, you can cap the frame rate either in the game's own config file, or that of ENB (if you use ENB, including just for its memory tweak). But don't do it in both, or you will probably get things moving in slow motion, as I learned the hard way.

If you plan to mod the game at all, I recommend following the applicable parts of the S.T.E.P. project's setup guide, as it is very well community-tested. It has good information on the various RAM and VRAM management tweaks that work (and ones that don't, and ones that get in each other's way).

I think if you're getting 60 fps, its good. I have read that skyrim gets buggy above 60 fps, mostly physics, giving rise to spinning stuffs n unrealistic physics on certain objects (no personal experiences on this one). I have a 2gb vram and 8gb ram too. And the textures are usually loaded in ram.

There is mod called Enbboost in nexus which can give better frame rates, it reduces ram usage. I have seen people arguing about it but my personal experience was good. And if you are finding physics problems you can try capping your fps at 60. And there is also a mod to see ram and vram usage inside skyrim - skyrim performance monitor.

Also if you are not playing in a 1080p monitor, I don't think 2k textures will give any better visual quality

VRAM usage will depend on display resolution, resolutions of textures, VSYNC (off, double, triple buffering in order of increasing VRAM demands), anti-aliasing and other features. If you're playing on a single monitor at 1080p, 1GB should be enough in most cases, whereas if you're gaming on 6 4k monitors you're going to need significantly more.

To summarize: can Skyrim use more than 2GB of VRAM? Yes. Does it do in your case? Depends on your configuration.

Some tools, such as GPU-Z, can measure VRAM usage. Learn more.