Looks like walnut to me … but actually the type of wood isn’t as important as matching the stain. I recently made my own dash out of walnut plywood, but the steering wheel I ended up with was mahogany and so I stained the walnut as mahogany and it turned out that walnut can look just like mahogany! LOL
You can buy small, thin pieces of walnut on eBay. I went the other way from you … I decided that I didn’t really need a glove box but would use a radio. Also, I thought it would be really hard to build a box into that area, given that there is all kinds of wiring back there. I built what “appears” to be a glove box using a small piece of solid walnut (stained mahogany) and simply hid the radio behind the lid. I figured it would be easier to do this than to build an actual box, but after actually doing it, I’m not so sure. Regardless, if you use thin walnut for your lid, you have to be very careful because it will split easily when you try to screw into it. But if you use thicker wood, it wouldn’t look good. And stay away from quadrant hinges unless you are some kind of master woodworker. I spent days with my Dremel, carving curved holes for the hinges to retract into.
Another idea would be to buy a VW Beetle glove box lid and cover it with veneer to match your existing dash. In 20-20 hindsight, that’s what I should have done.