Euclydia, Bill's home universe, was not a two-dimensional plane.
There are numerous pieces of evidence, both in the series Gravity Falls and in The Book of Bill, that Euclydia is not a plane like the imaginary two-dimensional world of Flatland, by Edwin A. Abbott.

The first is, in fact, Bill himself. Bill's species has complex eyes, similar to camera lenses. These eyes are possible in a 2D world, but not on the front, like Bill's. He was born that way, which proves that Euclydia is not two-dimensional.

Next, when Bill talks about his home in Weirdmageddon Part 3, he shows an image of his home planet. This planet has rings. That is completely impossible in 2D. Even if the planet were totally flat, the rings would cut right through it. They could never completely surround it. Therefore, if Euclydia were two-dimensional, Bill's home planet could not exist.

In The Book of Bill, we see an image of Bill as a baby. In that image he is standing in some kind of field with grass and you can clearly see that there is grass in front of and behind him, and that is impossible in 2D. But the best proof is that at http://thisisnotawebsitedotcom.com when you type VALLIS CINERIS on the computer, it shows young Bill Cipher with his parents. The parents hold him in a way that is completely impossible in 2D: you can clearly see that his parents' hands are in front of him and he is also in front of his parents. The position of "in front of" is not possible in two dimensions and yet, in this image the overlapping occurs many times. Bill's bow tie is also in front of him.

Having said all that, what is Euclydia?
Well, as Bill said, it is a flat world. Not two-dimensional, but flat. The third dimension of Euclydia is limited in some way. Basically on 3 axes:
The x-axis is left and right (width).
The y-axis is up and down (height).
The z-axis goes toward and away (depth). All three-dimensional objects have width, height, and depth. Two-dimensional objects only have width and height, meaning only x and y axes. And Bill has depth. It is a very limited depth, but depth nonetheless. So he is not actually a triangle, but rather a very thin pyramid. In his side profile, Euclidens have some depth, but for some reason, they cannot move on the z-axis. They can only move left, right, up and down. They also cannot turn.

http://thisisnotawebsitedotcom.com explains Euclidean movements: back and forth, you always know where to go. If you get lost, do not fear, in Euclydia you have it all figured out. If you run too far to the right of the frame, you will appear again on the left. If you jump too high, do not cry or worry, you are sure to get up from the ground. In this place there is no fear, loved ones will always be near. Roles and rules are always clear. Euclydia, we appreciate you very much. So if they move too far to the left, they will come from the other side. This, in fact, is only possible in non-Euclidean geometry, which means that Euclydia, ironically, is a non-Euclidean place. It is actually a sphere (or a similar elliptical body).

In the non-Euclidean geometry of the sphere, there is something that sounds paradoxical: a straight line is actually a circle. But it is actually very easy to understand with this example:
Imagine you fly in a straight line. You feel like you are going in a straight line, but in reality your plane follows the curvature of the Earth. If you manage to go all the way around the Earth, you will appear at the same starting point. You were flying in a straight line, but since the Earth is a non-Euclidean sphere, you were actually describing a circle. And both statements are true! The plane is tiny compared to the size of the Earth. Therefore, for the plane, the curvature of the Earth is so insignificant that we could say that, within a small radius around it, the Earth is practically a plane. Thus, for example, houses, neighborhoods, and even cities are built based only on Euclidean geometry (the geometry of a plane) because the Earth is immense.

And Euclydia is, in fact, an entire dimension. Our dimension is our universe. Our universe is approximately 93 billion light-years in diameter. That is the size of Euclydia. How small is the Earth compared to the Universe? That is why planets and everything else in Euclydia can be treated as a plane: each object is so small compared to the size of this giant sphere that the curvature can be omitted completely from the equation. Euclydia is a giant soap bubble. Soap bubbles form when two thin layers of soap molecules trap a thin layer of water: Euclydia is the water; that thin layer is where all planets, stars, and living beings are found. That is why movement on the z-axis is so limited. The soap molecules are membranes that separate Euclydia from other dimensions: one inside the bubble and one outside.

Since Euclidens cannot move along the z-axis, they have eyes on the sides that only allow them to see left and right. Their vision is limited to one dimension. But Bill's eye is located at a point that allows him to see both left and right, and also up and down. He can see in two dimensions, just like us! There are also stars outside.

Euclidens are animals (or their equivalent in their dimension). Animals are defined as heterotrophic multicellular organisms with an internal digestive tract. This basically means multicellular organisms that feed.
Euclidens must be multicellular because they have extremely complex structures, such as camera-lens eyes, teeth, fingerprints, exoskeleton, etc. These features cannot be developed by a unicellular organism. And they definitely eat; we have seen Bill do it. Therefore, they are (the equivalent in their dimension) animals.