Yep, mine did. Thought I had a bad sensor for a while until my tire shop confirmed the pressure in the spare was low. It was annoying that my spare was mounted with the valve pointed up, so I'd have to drop it just to check the pressure.
My spare does trigger the light when it's low, I don't have to drop my spare to either check it or add air because of the tire gauge and the air chuck I use with my compressor, the head on both is called "dual foot" or "dual head" they both look like the head on gauge and chuck at the links below.
I mount my spare upside down for this exact reason. A little difficult to get it to go up but you dont have to drop it to top it off to make the TPMS happy.