What objects do get loaded into GDDR? I currently have a 4K monitor and want to buy a new graphics card. My choice would either be the RTX 2080 or the RTX 2080 Ti. Now I need to know which one I really need. The Ti variant has 12 GB of GDDR6 instead of just 8 GB (and more shader cores of course, which makes the card faster)

My question is, what elements do actually get loaded into the GDDR6 memory? Is it only the textures or is there other stuff that needs to be loaded into the GPU memory? Does the memory consumption differ when comparing 1080p(HD) and 2160p(4K)? What if you would run a game at 4320p (8K)? People say that you need more GPU memory if you have a higher resolution, is this really true? And what things do you need to take into account when calculating the rough memory usage?
Typically you will need VRAM to store your frame buffer (memory used to store pixels for use with the display and for rendering). This would usually be 4 bytes per pixel so a 1080p frame buffer is 1920x1080x4 bytes (1 byte each for R,G,B components + 1 padding byte). Your card is going to use 2 of these for double buffering so it can use one for display output and the other for constructing the next frame. Any increase in resolution will require an increase in VRAM usage. If HDR is used then you're looking at 16 bytes per pixel so you can use floating point accuracy. Certain types of post-processing special effects may also need additional rendering buffers which are some division of the screen resolution so will consequently use more VRAM if using a higher resolution.

After that the majority of VRAM is going to be allocated for textures which for a typical game is the biggest user of VRAM. Then mesh data needs storing in VRAM which usually consists of vertices storing information like geometry position/normals/colours/transformation matrices/etc. Finally compiled shaders will need to occupy VRAM but they should be low down on memory usage compared to textures and mesh data. There will be other uses for VRAM too depending on how the game was engineered so this is not an exhaustive list of what VRAM is used for.

In short: Display resolution will change the amount of VRAM required but not by a large amount. For a video game most VRAM is from textures which are usually defined by the quality settings of the game and are not affected by display resolution. The 12gb card you mentioned will allow higher texture resolution but with your choice of 1080p or 4k display resolution making little difference in VRAM usage. However for 4k it makes sense to use the highest texture quality.