I am familiar with those connectors that use a sliding mechanism to clamp
the cable in place. Those have been around forever. It doesn't seem to be
designed like that. I've pushed and pulled gently in every direction and
if
it isn't one solid piece, I can't figure it out how to unlock it.

It's possible the cable is soldered to the board, with the "connector"
serving as a strain relief. However, soldering the cable would make the unit
difficult to repair. (Hmmm...)

Do you have a good magnifier? A careful nosing around might reveal something
not apparent in the photo (which, by the way, is of excellent quality).