Is it possible to fit a SPD pedal and cleat setup on my SPD SL shoe (SH-R131-S)?
The answer to why, is to save the expenses for a new shoe.
But even if it's possible. Is it a good idea to mix them?
Yes, it is possible with a suitable adapter. eg the Shimano SM-SH85, or the Crank Brothers 3-hole cleat adapter. These bolt onto 3 hole (road) shoes, and provide 2 holes to bolt on SPD cleats or similar.
The main disadvantage is the thickness of the adapter. This will make the cleat stick out further from the shoe, so it will be closer to the ground. If your shoes don't have much tread, the cleat may be touching the ground. So it may be very slippy if walking on a smooth floor. And the cleats will wear out faster, especially if walking across stones or rocks. Also, road shoes can be very stiff, so they may be uncomfortable for walking in.
Proper SPD shoes (designed for mountain biking or touring etc), have the cleats more recessed, so they don't touch the ground. And they are usually less stiff and have more grip, so better for walking.