1 x CR2032: 2.98v (off) -> 2.82v (on)
2 x CR2016: 5.98v (off) -> 2.63v (on)
Interesting! Why does it do this?
Because when the circuit is trying to pull more current than the battery internal resistance allows, you get voltage droop.
The real question is what the LED specs allow for max current, which may also depend on how well it is heatsunk (or in this case, heatsinking an encapsulated bi-lead LED can depend upon a short LED cathode lead soldered to a good amount (thickness) of copper wire or PCB layer.
It could be that the designers knew the 2 x series CR2016 power supply was inherently current limiting so they didn't (need to) design for higher LED power, so driving the LED at higher current could overheat it.
To clarify it is not a "6V LED", it's (presumably white) 3.(n)V LED but the tiny coin cells are acting as a current limit, plus as already stated, older cells already at a lower voltage.
If you power this for long from the single CR2032 cell, you might find that the LED more quickly gets dim and a significant % of the CR2032's capacity is wasted because the partially discharged voltage plus voltage droop is below the forward voltage of the LED. CR2032 cells aren't
"good" for much more than 20mA if that, driving a white LED.