Honeywell thermostat doesn't seem to be connected to standby generator
There is likely a 24V transformer somewhere that connects to both the thermostat and furnace. From what you described, my hunch is that the thermostat is on a separate circuit, not together with the furnace.

It could be a wall wart like this one or it could be a direct wired transformer like this one, in which case it may be mounted in or on the furnace (though then it would normally pull power from the same circuit), or it could be mounted to a junction box nearby.

Once you find it, you need to either move the circuit to the generator panel, or move the transformer to a different circuit that is already on the generator panel.