I got some help from folks here identifying this as a gas unit heater.
I'm wondering if it's safe to replace the thermostat with a smart thermostat, and if there are any caveats, E.G. minimum on and cool down settings. (Should I be wary of regular home heating advice given this kind of heater, or am I over worrying?)
The main reason I want to get a new thermostat is that this thing currently reaches target temperature, turns off, then kicks back on when it falls 1 F (or not even, it seems), which is way too often for my poorly insulated apartment. I'd prefer something like 5 F.