The relation of the lit side of the moon and the sun

I'll assume you mean the lit side of the Moon (this wasn't clear from the original question, but I've edited it).

Suppose there's a crescent Moon visible in the sky during the day. Plot a straight line between the tips of the crescent, then plot a perpendicular straight line across its center across the sky. Intuitively, that line should pass through the Sun. You're saying it doesn't, and you're asking why.

In fact, that line does pass through the Sun, and what you're seeing is an optical illusion.

If you imagine the sky as a perfect hemisphere, with you at the center, the geometry will match up. But we we tend to imagine it as a flattened dome, probably because the horizon is obviously far away, but we have no reference points for the zenith so we imagine it being closer. This is (part of) the source of the "Moon illusion", where the Moon appears bigger when it's close to the horizon. In fact, it's very nearly the same angular size, but since we imagine it being farther away it must be "bigger".

The same mention distortion of the sky's geometry makes it look like the axis of the Moon's lit side doesn't point directly to the Sun.

Next time you see this, try holding up a straight rod intersecting both the Sun and the Moon. I think you'll find that the rod passes directly across the axis of the Moon's lit side.