I took a second look at the picture of your exchanger .... it is wired improperly just so you know.
The 14-2 wiring from the household system should be clamped into the power distribution box unless the wiring is fully enclosed. The outer white cover should be in the clamp. Here are picture of the 2 most common types of clamping.
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The above is a 240 volt circuit, a nut would go on the threads to hold the clamp in place to the box.
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This is one with an internal clamp for use in walls.
The clamping is done for safety reasons, putting the white cover in the clamp is to keep the inner wiring covers from being damaged and shorting on the box.
I appreciate the concern, but it looks worse than it is.
The gray wire *is* clamped as per one would expect; it's a power cord plugged into a receptacle below the air exchanger.
The thing that's throwing you, and understandably, is it looks like the romex wire is what's wired improperly. The romex is actually the controller wire that leads to my humidistat that controls the air exchanger.
This is actually the second air exchanger I've had. The first one was installed in 1994, replaced in 2016 or 2017, can't recall for sure. Got 22+ years out of the first one, nothing to complain there.
The original one had the control (humidistat) controlling the 120v current to the exchanger; in other words, the humidistat was the "on-off" switch.
The newer one uses 24-volt to run to a controller. I really liked the way the humidistat controls indoor air quality; the new one is designed to do it by time (x percent of the time the air exchanger runs), but the humidistat just bumps on the air exchanger when humidity.
When I got it, I asked the plumber who installed it if I coudln't just run the 24-v through the humidistat using the old wiring. Certainly it was robust enough.

I didn't want to run new wire if the old would do. Answer? Why not?
So what you see is romex wire for a 24-v circuit, not main power.
We built the house in 1991/92. I built it to be as close to superinsulated as possible, but the bank would not let me built a house without a furnace, and part of how you fund the extra insulation is no furnace. You don't need one in a superinsulated house.
So I only got about 2/3 of the way there. I personally insulated the house myself, at night after my day job. I sealed that house to within an inch of its life. Turns out, up to half the heat loss in a house is due to air infiltration, and sealing the house tight eliminates that.
But--you have to ventilate a house, and that's why the air exchanger (I'm sure you know this, it's for others who may not). Otherwise, indoor air pollution, way too high humidity, bad juju. However, I want to control that instead of it just happening as air infiltrates through cracks and openings and such.
And I do that by a humidistat.
FWIW: Here I am in Wisconsin. We had the polar vortex about 6 weeks ago--we were down to about -30 here at night, wind chills down to -60. I just got my utility bill for that stretch. I heat with natural gas. During the coldest stretch in decades, my monthly nat gas bill for heating? $79. That is for a house about 2500-3000 sq ft depending on how you want to count the finished basement.
Not bad. And a lot of the credit for that goes to the air exchanger.