Only a handful of acft were used by navy and army :
Mitsubishi Ki-15=C5M and another one I can't remember.
But they used many weapons and engines together even if called differently
(Nakajima NK1 Sakae = Ha-25 etc)
By the end of the war, at least engine numbering was standardized.

Did the Imperial Japanese Army Air Force and the Imperial Japanese Navy
Air
Force operate any of the same make of aircraft? It seems to me that that
at
least with some combat aircraft the two services might have economically
ordered from the same production lines and saved some money and opened up
some
factories for producing more of other types of aircraft?

They should have and the war would have probably lasted longer.. 'hopefully'
rivalry was huge btw navy and army.