Homelink had to use remote pairing to determine the remote control frequency operating band, Car2U cycles through all the ones it supports instead. That's why they eliminated the "copying the door remote" step but it's really in the interest of cutting equipment costs since it doesn't need to receive anything anymore.

My speculation (hard for me to troubleshoot these things without seeing and touching) and barring the user error factor entirely, is that the Car2U transmitter is "rolling" every channel at the same time instead of independently. When you "pair" the door operator to the remote, it's not a 2-way handshake like Bluetooth - it's more like telling the receiver "start responding to this pattern when you hear it" and the transmitter is putting out the same pattern for all the buttons for some reason.

Did this start recently or has it been like that for a while, or did you just use this feature for the first time?

If all else fails, a workaround to the problem would be to clear the codes off one operator and purchase a receiver with the dry relay contacts that go in parallel to the white and red control wire pair for the wall-mounted switch.
A product along the lines of this:
Sears: Craftsman Universal Receiver 315MHz