As far as I know, the late Miocene volcano preserved at Sibley Volcanic Reserve doesn’t have a name, so no one will mind if I dub it Sibley Volcano. This is what it’s mostly made of, a dark basaltic lava as exposed in the rock face behind the water tank.
The bullseye pattern is the result of spheroidal weathering. A thick body of fresh, solid rock generally is quickly cracked by sets of joints (a joint is fault that has not yet moved, or a fault is a joint with displacement on it). Joints let groundwater, among other things, into the rock. As the groundwater does its thing on the rock, it works its way into the jointed blocks and, to paraphrase an old Grateful Deadhead saying, the smaller they grow the rounder they get. Spheroidal weathering.
The wonderful rounded granite forms of Joshua Tree National Park result from the same process. They were gradually exposed by erosion and, being coarse-grained granite, they don’t display a dramatic onion-skin structure. This was abruptly exposed by quarrying. It’s a miniature version of the exfoliation domes in the Sierra Nevada granites.