That does not make any sense because hole plating is a chemical process not a mechanical one, the process will not care if the hole is circular or not. And this is not the same as edge plating
I think the issue here is that for a certain size and shape of the hole, it isn't uncommon for the process to have to be done during a milling step. Milling often comes after the plating step. I.E., All the holes are drilled, the board is plated, etched, solder mask, silkscreen, and surface finishing added, and only then is milling done.
If you have holes that need to be plated that are incompatible with the drilling machinery, you either have to add a milling step before plating or a plating step after milling. I suspect the confusion here is that the "plating step after milling" is when edge plating would occur so some people call it 'edge plating' if it gets done at that step. Note that all of the above is manufacturer-specific so what can be plated without additional cost/steps is going to vary depending on who you get to make boards. Even my low-cost manufacturer can do 'elongated holes' up to a certain dimension, but they can't do anything more than that if you want it plated.
For the original poster's application, I'd probably just not worry about having the hole plated. Put a pad top and bottom and make sure that I had enough vias such that the circuit itself wasn't depending on the connector making the connection from top to bottom. Part of the reason why I would do this is that I know that a plated mounting hole on a multilayer board is just asking for problems, as the mounting screw can cause various types of damage which is worse with a plated hole (such as fod, or cracked barrels, etc). So, for mounting holes that need to be connected to the chassis through the screw, generally, the hole itself isn't plated. Instead, a ring of vias is placed around the hole with copper being on top and bottom of the board - but no plated barrel in the hole. Ideally the vias would be outside the screw head area to prevent them from being damaged as well.
For the OP, mounting the connectors through the board will result in similar pressures/potential damages as you would see with a mounting hole which is why I'd just do the same thing as I'd do for a grounded mounting hole.