The joint you're referring to is normally known as a "dovetail key" or "butterfly joint".
The usage you see here is not typical. Normally this would be used to keep a wider board/slab from "checking", or splitting along the grain from the end of the board. As such, the key would be oriented perpendicular to the grain of the larger board.
There are many ways to make the key, but the easiest is probably just to lay it out with a bevel gauge and cut the waste out with a hand saw. You'll want to leave it as a part of a larger board to allow easier handling and crosscut it free as the last step.
To inlay the key I would just place it where I want it, trace around it with a pencil (or, better yet, a marking knife) and then chisel out the waste.
I would start chiseling carefully around the edges, alternating paring down (with hand pressure at first) and then pushing in almost horizontally (bevel up) to your pare line to pop out some waste. This is similar to how you'd start a "first class cut" with a hand saw. Repeat this deeper and deeper until your edges are well established and then hog out the center waste with your chisel (bevel down) and a mallet.
Repeat establishing edges and hogging out waste until you're at your desired depth. (Make sure the depth of the mortise is less than the thickness of the key!)
Alternatively, you can hog out most of the waste with a router and a plunge bit and then clean up the edges with a chisel. Just be very careful not to let the router get away from you. Practice on some scrap first to get a feel for how the router handles freehand.
It may help inserting the key if you put a slight bevel on the bottom edges.
Once you glue it in the key should be slightly proud of the surface. Flush it up with a hand plane (or a scraper/chisel/router/belt sander/whatever you prefer).