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enter image description here

Why does the texture skew when you subdivide the plane once?

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    $\begingroup$ If you look carefully the texture is also deformed with 2 subdivisions compared to the original square plane. The original square plane has each or its 4 vertices mapped to 4 corners of the UV space. When you use the subdivision modifier level 1 its adding 4 more vertices to the square plane forming an octagon shape in the 3D viewport. This octagon has all of its vertices still mapped to the outer edges of the UV space (square). This causes the image to be deformed. 2 edges of the octagon are mapped to a single edge of the UV square. Apply the modifier then UV unwrap (map) the mesh again. $\endgroup$ Jun 24, 2020 at 11:49
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    $\begingroup$ Interesting. Coincidentally recently linked this blender.stackexchange.com/a/93180/15543 to an answer. Seeing the subd edges might shed more light. Nevermind moony has done that. $\endgroup$
    – batFINGER
    Jun 24, 2020 at 11:49

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Each quad face is always made of 2 triangles in 3D softwares. I guess the result you see has to do with the way Blender will triangulate each of the 4 faces it creates when it subdivides.

Here I use a grid as image texture to make it more understandable. On the left, the original object, on the right, subdivided once. Here is how Blender triangulates:

enter image description here

If I select my original object, cut edges the same way to make only triangles, and turn the outline into a circle, I'll have the same result:

enter image description here

It would work better that way:

enter image description here

With a higher subdivision, the problem tends to be less noticeable.

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