Anyway, here are a couple of other 1/8" chisels we have sculling about the w'shop, which seem to me to narrow down the options. At the bottom we have our trapezoid by James Howarth; above that my old man's Wm. Marples bevel-edged firmer, which is actually a hair under 1/8"; finally a OVB mortise by Isaac Greaves. The latter some of you might fancy to call a guinea-pig sticker, I suppose. If you were that way inclined. I'll forbear from comment if you do...
So our un-named chisel style could, I suppose, be a straight-sided firmer. It's not impossible they'd be made so noticeably different from the bevel-edged, just for the sake of strength. But personally I did immediately warm to Joel's thought that they were used in caning chairs; it rings true. Of course you might argue we see them so regularly, this seems unlikely - but then cane seats were a lot more prevalent in days of yore (Almost as bad a phrase as "Yesteryear" - my apologies). However, the clinching argument for their widespread appearance is probably best summed up by many of the comments you all left as in any other way.
Viz: Every woodworker can think of a task where a handy narrow chisel like that would have/will come in handy ;)