I can connect data between cells but only the values match. How can I connect cells between different sheets so that if I change the fill color of a cell on sheet 1 it will also change the color on sheet 2?

Manual color choices are not transferable to other cells via formula.

Instead of manually employing color changes, you could create Conditional Formatting rules to change the colors for you based on cell values. Then you could apply those same rules to the other linked cells, thus both cells would change together.