The images you linked are raster images, aliasing would have very little effect on them.
It is true that Godot doesn't have aliasing in 2D, but aliasing mostly effect vectors, sampling the sharp edges to get a softer edge.
For your own images there is a few design principles you need to follow for Filtering to work; filtering is what you are looking for.
1.) A small empty boarder needs to be placed around every image, 2 transparent pixels on each side should be perfect.
2.) Mips and Filtering only work on images of power2 that is 64x64 or 32x512 etc. X and Y doesn't have to be the same.
List [1, 2, 4, 8 ,16 ,32 ,64 ,128, 256, 512, 1024, 2048, 4096 etc.]
3.) If you have buttons that are odd sizes like 100*50 pixels you will need to atlas them on a sheet that follows the power2 rule.
You can often fit the whole GUI on a single 2048x2048 texture atlas; saving a lot of memory and loading time.
You will need to make manual "TextureRegions" for each button. Because the Godot AutoSlice isn't going to keep the 2 transparent pixels you need.
I find the Godot TileMap tool works for this.
For the text: Under DynamicFont->settings->MipMaps and Use Filter here.
Fonts follow the power2 rule so enabling them will work without any problem.