1) I'd check the behavior with another trigger source that leaves the MIDI module out of it as Ville suggested.  It'll help isolate the problem source.
2) one theory is that with a single note, the voltage actually starts at 0v but with many notes voltage hasn't quite dispersed when the envelope is retriggered with successive notes.  You many lose a bit of voltage on the triggers as you are connecting between modules/ patched between separately powered circuits.
3) Are you connected simultaneously to the BUS from any of the modules involved   Perhaps isolating them may help.  (variation of #2 in that you may lose voltage to the BUS on the first note)
4) The sample rates of the digital chips / internal clocks involved but it could also be a minor contributor. Sound happens fast enough when you press a key so you don't notice it on single notes.  With a secondary notes, you notice the envelope doesn't quite respond the same way after a NoteOff/NoteOn so you can tell via the pitch/CV change. All-in-one synths tend to have a system that has synced timings.