You only need vertical & horizontal coordinates in the same units when the calculation actually requires that both are in the same units. For example, calculations of slope (rise/run or vertical/horizontal) or aspect which is derived from slope.
Other examples of when you might need to reproject are when you need projected (X, Y cartesian), not geographic (lon, lat degrees) horizontal units for the calculation, such as density (values per unit area) or cost/path/euclidean distance. Obviously here the raster values are not vertical or elevation values and so don't (and can't) be in in the same units
If you're just draping a polygon or raster region over raster values to get some summary stats, you don't need to reproject. Indeed by reprojecting, you are introducing an additional source of error.