I sometimes see structures that are 'almost symmetric' - like in, 'if only one changed this one thing/few things, then it would be symmetric'; this had lead me to think about whether there is any meaningful way in which one could quantify symmetry in general - and which form it might take? Something akin to a 'stress tensor', perhaps?
What I'm imagining is something that would allow us to say 'A is more/less symmetric than B', possibly adding 'along ....'.
Sure, so suppose we have a shape. Draw a line that cuts the shape into two parts. If the two parts are reflections of each other across the line (think of it as a mirror), then that line is said to be one of symmetry.
For instance, a circle has $ \infty$ lines of symmetry (any line passing through its center). A square has $4$, two of its diagonals and two lines (vertical and horizontal) passing through the centre and parallel to two sides. But any randomly-drawn messy shape probably does not have any line of symmetry.