The wood color looks kinna like cherry. The wood might be cherry. Cherry darkens over time. It doesn't appear to be walnut or mahogany, but with pics it's hard to tell.
If you could get some scraps of cherry, you could
1) test some finishes,
2) test some stains if need be or
3) possibly another option is to faux weather it, then apply some mineral spirits (giving it the wet look). The wet look will show you how it will likely appear when a clear finish is applied.
Using wood other than cherry: Find or blend a stain to match the old wood.
To faux weather any wood, soak some steel wool (SOS soap pad size) in 2 cups of vinegar for several days.... until the steel wool dissolves (stir occasionally). Use this solution to "paint" the cherry scraps.... wet the wood pretty good..... , then allow to dry. You'll see how it darkens or grays the wood. In warm sunny weather it will dry in 15-20 minutes. Once darkened or grayed, wet it with water or mineral spirits to see whether the color matches your old wood. If it needs to be darker, apply another painting of the steel wool solution. Do some testing. Faux weathering cherry might give it the aged/darkened patina look of old cherry. Again, I'm not absolutely sure your wood is cherry.