This Starrett 154E parallel and Goodell Pratt 1218 inside mic were in a flat with some other measuring tools. The rods do not go with the mic but I carried them home so I'd get my $4 worth. I don't remember what those brass thingees on the 6 inch scale are but stair gauge is in the back of my mind.

I've always called those brass thingy's Stair Gauges, don't know what the correct name is.

I'm with both of you as to the name.  So are the manufacturers I found listed on Amazon.  Another, perhaps somewhat antiquated but neat* term is "stair buttons."

*That was a touch redundant: the antiquated terms are almost always neat-o.

Starrett No. 470 Stair Gage Fixtures. used with a carpenter square for lay out
Bob