Why can't I exactly match the same points on different VFR sectional charts?
This is by no means specific to aviation charts. Most likely it's simply general property of local maps.

Maps are flat, but earth surface is (approximately) spherical. So to represent ground features on the flat map a projection is needed and this projection can't preserve all relations—angles, distances and areas—precisely. So there are many different projections that distort different properties.

Small area maps are usually drawn in azimuthal projection with the central point in the centre (or significant point near the centre) of the mapped area. While such projection does not preserve exactly any properties out of the centre, the distortion grows with distance from the centre, so for small areas it's appropriate.

This however means, that each section uses different projection and that near the edges the maps will be distorted each in a slightly different way. That's why the maps can't be easily laid out to form a bigger map. You would have to lay them out on a sphere.

You could try using some software for stitching panoramatic photos (here is one such you can use for free). That should be able to apply suitable transformation to make the maps match. At the cost of increasing the distortion.