How would I go about identifying and matching this wood stain color?
Well first off: that's not oak. It's larch or southern yellow pine (maybe even hemlock) from the look of it. As for color, that looks very near to the natural color of softwood after some yellowing from age but its hard to say. It appears to be production grade furniture (although of a fairly high quality) which means the colorant(stain), if any, is a proprietary blend mixed specially for the manufacturer by a finish supplier (i.e. Sherwin Williams) so trying to track it down will be difficult or impossible. But don't desperate, most furniture makers have a "natural" color blend and most of them are very similar.