It is "Axial" because the axis of each cylinder is aligned with the axis of the output/crank shaft.
This type of engines are also called 'barrel' and 'Z-crank' engines. The former refers to the cylindrical shape of the Cylinder Group whilst the latter alludes to the shape of the Crankshaft.
The Barrel shape is a result of the pistons being spaced evenly around the central Crankshaft and aligned parallel to the Crankshaft axis. The 'Z' in the crank provides the journal surfaces upon which the combustion loads act to provide the driving torque of the engine.
The uniqueness of the DUKE Engine is the combining of these two motions in a counter-rotating configuration which results mechanical and performance advantages according to Duke.
Source: http://www.dukeengines.com