Not the same thing per say. The button on your mouse most likely changed the way the sensor read the DPI. Gaming mice have a very High DPI that they can read or cut to change how sensitive they are. This is all done with the Hardware.
A standard mouse has a fixed (usually low) DPI it will read at. You can use Software to adjust the sensitivity, but it wouldn't be quite the same, and assigning it to a button would require some third party software/magic.