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I doubt it's a mistake, artistic licence at most. I watch SGA's videos and I think the titlecard artist (EDIT: I don't know who that is) knew that the hyperbolic plane is more like the opposite of a sphere.
But simon_clarkstone is right in that the hyperbolic plane would not actually LOOK like plane to us. In Euclidean geometry, a plane will always have a viewing angle of 180 degrees, but this is not true in hyperbolic geometry. When you are at a distance of x above the plane, the viewing angle is twice the angle of parallelism, or 4*arctan(exp(x)).
So, imagine the PC's eyes are 1.8 m above the ground. That is 0.6 of absolute unit, and inputting this numbers gives us tha viewing angle of the plane 115.034 degrees.
Now, if you are on Earth: how high would you have to go to see it under this same angle?
The viewing angle of Earth in Euclidean geometry from height h above its surface is 2*arccos(r/r+h). To see Earth under this same angle, you'd have to be about 4800 km up!
Now, if you were to see a sphere under the same angle at the same distance (1.8 m), what would be its radius?
About 2.4 m. So the HyperRogue plane looks similar to a sphere of radius 2.4 m when you stand on it :)
lengths must be calculated not just measured. lines don't look straight lines .
the hyperrug and the hyperbolic paraboloid (and other surfaces as well) have saddle points and have a negative curvature, lines do look straight, length can just be measured with a measuring stick polygonssthat are the same size will also look to have the same size, see https://en.wikipedia.org/wiki/Hyperbolic_geometry#Hyperbolic_plane_geometry_as_the_geometry_of_saddle_surfaces
To visualize what Fulgur14 said  go to the Ivory Tower, get some meters above the ground, and you can see how the ground level looks  it does not cover the bottom half of your vision as on Earth, but just a small angle, the further you are, the smaller it is. In three dimensions this is similar, so If you were looking at HyperRogue's plane from above, you would see a disk of that size. So it seems that the art is actually not that inaccurate :)
As others have said, a plane in hyperbolic 3space looks similar to a globe from farther away, just as "lines" (or geodesics) look like curves from a distance. I think it'd look a lot different than in the illustration mentioned by wilemien, and that "globe" would appear much larger in scale, where everything at the edge is dense and becomes infinitely small, but the main idea of depicting it similar to a globe is accurate.
And here is a direct link to the art in question: https://i.ytimg.com/vi/DmrjxGDHT0Y/maxresdefault.jpg
If I identified the characters / objects etc. in the picture correctly, they are:
Monsters: Cultist (or Desert Man?), Slime Beast, Tentacle, Running Dog, Sandworm, Hedgehog Warrior, Fire Fairy, Demon, Shark. Lands could be Icy Land and Dry Forest, with a hint of a Desert area to the left. Objects are probably walls.
It turns out that:
 If the base level in HyperRogue is actually a plane (that is, B=0), we will see it in the Klein model. This is actually quite obvious: straight lines on our plane are actually straight, so they will have to look straight to our eyes. The value of C is irrelevant for the shape  only for the angular size.
 If B=C, we will see it in the Poincaré model. Since L is an equidistant surface, straight lines on it are not actually the shortest paths if we are allowed to leave L  we get shorter lines by moving closer to P, so lines will look curved towards the center.
 In general, L will look like if we set the projection parameter in HyperRogue to A=(cosh(C)*sinh(B))/((cosh(B)*sinh(C)).
The new 3D mode in HyperRogue (obtained by changing the "wall display mode" in the basic config, see the bottom of http://www.roguetemple.com/z/hyper/gallery.php ) effectively works by changing the projection parameter depending on the height, so it gives an accurate view of HyperRogue's 3D world from above, for some values of the parameters.
Note: This was actually quite surprising to me. In the Halloween minigame played on the sphere, we have 3D walls of the holes, and they are created simply by moving the point toward the center of the sphere. The 3D mode in the hyperbolic plane was inspired by this, and it does the same thing, but for the hyperboloid in the Minkowski space: the standard view in HyperRogue is obtained by viewing the hyperboloid from the point A units away from the center of the hyperboloid C; now, if G is one of the bottom vertices of a wall, the corresponding top vertex is the point C + (GC) * 1.25 (this value will be configurable in the future); note that the perspective has to act a bit weird here, because we see the furthest point, not the closest one. Thus, top faces of the walls are effectively viewed as if the projection parameter was 0.8 A.
It uses the model described in the last post, and shows it from the point of view of a bird flying over the world. It looks quite close to Memoski's graphic :)
The video is made in HyperRogue  by changing the "camera rotation" parameter (menu>special display modes>3D configuration>camera rotation) and other 3D parameters.