why is square color important in chess?

On a board of one color, all the rules would still be applicable, and the game play would be no different. It's just a visual aid.

The queen always starts on her own color.....

If you can dominate the squares of one colour you will likely have a good position.

If you switch the king and queen placement for both sides at the start of the game it's the same as letting black move first. Nothing is changed, the colors just appear to be swapped.

Is the position less dominating if the colors were the same?

Ok, but I guess I see what you guys are saying. Maybe this is what the OP was asking for.

Your question could mean about three different things. Using color is just a simpler way of describing the geometrical situation whereby there exist two natural sets of diagonals on any grid that never intersect, which has implications regarding piece coordination, piece utility, holes, and so on. Bishop colors/diagonals have implications beyond the immediate. As an extreme example, if a chessboard were 9x9 instead of 8x8 then the color (= diagonal set membership) of the bishop would determine whether a bishop-knight mate were possible with those pieces on that board at all.

https://en.wikipedia.org/wiki/White_and_Black_in_chess

I can only speculate, but I'd say that the colors in the squares are more for conveniency and aesthetics than anything else. As said above, the colors probably facilitate geometry visualization (and maybe memorization) on the board.

What really matters are the coordinates, not color.

That's how chess boards used to be. It was just a grid of squares. Just look at go and xiangqi.

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Go and Xiangqi are played on the lines and intersections, the square colors are irrelevant. They're basically just the "space inbetween the lines".

You could still see the squares on the chessboard if the squares were outlined with a contrasting color.