What is a "unique" mirror line of symmetry? For example why does an equilateral triangle have three mirror lines but only one "unique"mirror line of symmetry?
Some Googling turned up this excerpt from what appears to be the transcript of a Coursera lecture on this topic:
If we take a hexagon then evidently, there are mirror lines but we can draw them in two different ways. We can either draw mirror lines which are passing through the vertex of the hexagon and you would have three of those. Or you can draw those mirror lines passing through the face of the hexagon and there are three of those [...] so that's why we would say they are mirror lines of only two types.
So insofar as this topic is concerned - since it clearly is not emphasizing a rigorous presentation of definitions - it seems that you should consider a "unique" mirror line to refer to the equivalence classes of mirror lines, up to congruence of the mirrored halves.
Because the shape when cut using mirror lines are different. For example, the shape when you use vertical and horizontal mirror lines, the shape of clover is the same (2 petals of clover). When you use the diagonal mirror lines, the shape of clover is again the same but different from the horizontal and vertical mirror lines (1 petal +2*0.5 petal). Hope this helps