It is highly unlikely that your raster area will ever exactly match your vector area by the very nature of rasters.
You vector polygon boundary will cross through your raster cells. When extracting by polygon, Arc uses the center of the cell to determine if a cell is to be included in the output. If the center of the cell is inside the polygon, the cell will be included and, if the cell center is outside, the cell will be excluded. The raster cells are not sub-divided. Their size if fixed by the raster resolution. The more convoluted your boundary, the more edge cases there will be.
There is therefore no way to guarantee that the raster area will exactly match the vector area (even when rasterizing a vector polygon). However, you can bring the results closer by using a finer resolution of raster.