Sorry, Jupiter was a Roman god. And he could spell it right I imagine.
No, the identification is correct (though you're rght, the name is spelled wrong, by our typical English standard). "Jupiter of Doliche" indicates that this god was a local syncretisation of an Anatolian god with Jupiter, in much the same way that Zeus/Jupiter would be syncretised with any other Eastern god. The battle axe symbol probably does indicate some syncretism with Mithraic concepts, though the lack of the Phrygian cap would seem to suggest that this figure is not Mithra himself.
That was my first thought.
Mithras was a soldier's god during Roman times. It was also speculated that the historical Arthur (Arturo the Bear, yes he did exist) was a follower of Mithras.
Oh my gosh, you sound super smart!