I think the answer is a qualified yes, at least at DC. I haven’t checked anything at AC just yet.
Results so far:
I’ve picked 5 ULN2003 from the junk bin - all different lots, and from at least 2 different manufacturers. At 1mA, diode forward voltages match +/-0.5mV across the chip, at room temp, in all 5 of them.
I’ve ran those chips through a range of currents, from 1uA to 8mA, and 5 out of 7 diodes usually match within +/-0.1mV of each other. Tracking is almost an order of magnitude better.
Here's the static (DC) raw data for one of the chips, and the plots of it, including a black reference line (manually fitted). Highest precision is achieved within two orders of magnitude. Towards lower current, the mechanical stresses become an issue, and my test setup introduces some repeatability error as well since it's thermally rather crude.
I’m not disappointed at all!
It looks like that the diodes closest to the sides of the die run at a slightly different temperature than the ones towards the middle. So a rule of thumb could be that a ULN200x has 5 diodes that are best matched - channels 2 through 6. My application needs three, so I’ll use channels 3,4,5 for that.
The diodes track well across all channels, but there's a slight channel-to-channel matching error - well under 1%, and mostly around 0.1%. That's an excellent result for such a basic chip, I think.
The voltages were measured with a DVM with input impedance of 10 Gohm.
I’ve also scoured the schematics of a variety of low- and high-side bipolar driver chips for any B-E junctions that could be accessed externally - rather than just the diodes (if present). Nothing promising yet.
For ULN2003, I've also measured the output V-I curves when all channels are turned on with 5V at the inputs. The voltage drop across the output transistor does not follow the exponential law. This is not unexpected, since the transistors are saturated and act like resistors, more-or-less.
I’ll be coming back to this answer and editing stuff in as I find out more.