Well, for one thing US impeachment is something the House does and which leads to a trial in the Senate. And is preceded by extensive quasi-legal hearings in the House (judicial committee) as well. It's a specific procedure to try people who enjoy immunity from normal criminal proceedings.
This is a mechanism for the Governor to remove somebody from a Commission, with Senatorial approval. There was no semblance of even a kangaroo trial. It's really, really not what the word "impeachment" was invented for (not by any American, need I add?) The only thing that vaguely resembles an impeachment is the line about this being only in case of (do I need up the exact wording? Nah) - specifically where they're as hazily defined as in the US Constitution. Which, obviously, opens the floodgates to judicial interpretation.
Sure, there are differences in the details (although not all of these details are constitutionally mandated). I'd quibble with your description of federal impeachments, however - the purpose is not to try officials who enjoy immunity. They generally do not enjoy immunity from prosecution, and impeachment is not a substitute for prosecution, as that can occur independent of any impeachment. The purpose is to remove the official from office, full stop.
In any case, it's not clear to me that these procedural differences mean that the act of impeachment by Congress is non-justiciable whereas the act in question by the AZ Governor & Senate is. Both acts are exercises of explicit powers granted to the non-judicial branches by a constitution, and are actions, not laws: no part of removal from office requires judicial action, so it is not at all obvious that this should be subject to judicial interpretation as you claim, nor that they differ in their justiciability. Both powers are granted under some vague threshold condition, but if Congress is the only body whose interpretation of that threshold matters in the federal case, it stands to reason that the AZ Gov/Senate's interpretation of that similar threshold is the only thing that matters here. Courts do not always get the last word.
