My fith front tire went flat due to a nail when parking on the street. Had AAA change the wheel and replace with the spare. After replacement with spare dashboard shows light indicating front right tire is flat. But the spare wheel has a brand new tire with full pressure. I suspect it does not have the tire sensor inside which the other wheel did. Is that why the dashboard still shows flat tire? I am sure you guys know the answer. Anyone?
The right front tire IS still flat and mounted where your spare was. Unless you TELL the car where is tire is located, it won’t know, so it still thinks the RF tire is on the RF. I don’t know if your spare has a sensor or not. If it is one of those temporary spares, likely not, You’ll need fix the flat and swap it for that temp spare THEN the light will go out.
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Nice catch, Mustang. I agree. 
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Check your owner’s manual, you may be able to correct the tire sensor display yourself.
Inflating (repairing/replacing) the flat tire is the solution. If the tire in the spare location is damaged and deflated the light should be on to alert you.
Without compact spare tire
The spare tire is also equipped with the tire pressure warning valve and
transmitter. The tire pressure warning light will turn on if the tire inflation
pressure of the spare tire is low. If a tire goes flat, even though the flat
tire is replaced with the spare tire, the tire pressure warning light does
not turn off. Replace the spare tire with the repaired tire and adjust the
proper tire inflation pressure. The tire pressure warning light will turn off
after a few minutes.
On my brother’s tires, the spare tire is full size, mounted on a regular alloy rim, same as all the others. It has a sensor, and the module monitors all 5 sensors
I don’t know what year yours is, that might be helpful