Why do they have to be geometries of their own kind? For example, instead of talking about lines in spherical geometry, can’t we just imagine circles on a literal sphere in Euclidean space?
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Look at a map of the earth. Draw a straight line between say London and Los Angeles. Is that the shortest path? It’s not, because the surface of the Earth is not a Euclidean surface and a map of the Earth (which is technically Euclidean) is a distortion of the actual Earth. At local distances the earth is flat enough for Euclidean geometry to be good enough but it doesn’t take that far for curvature to start to matter.
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