I can't speak for Mazda...
But on GM vehicles with an ambient temp reading on the mirror/display, they use a crazy pre-programmed set of code that estimates/calculates the "true" ambient air temp based on how long the car has been running(or not running), how fast it is going, it compares the ambient air temp sensor vs the intake air temp sensor.
For instance... If you've just shut your car down and come out and start it up 20 min later... normally the AAT sensor would read artificially high because of heat soak of the engine and radiator inflating the reading. It calculates the amount of time since the last time the car was shut down and uses the coolant temp vs ambient air temp(when it was shut down) readings in a table to display the correct "calculated" exterior temp on the display in the mirror.
It's pretty fawkin' wild... FWIW I wouldn't worry much about what the temp display reads anyway.
I'll add that if you're missing your hood "gaskets" and stuff, your stock intake won't be taking air in from outside, it will be taking it in from under the hood of the car because heat rises... meaning all the engine bay heat is raising straight upwards and out of the front of the hood, where it is recycled back into your intake.
at idle.
If it's still reading high while driving then it sounds like your sensor may be reading wrong, or you have a wire going to/from it with high resistance(possibly... I'd have to look up if higher resistance would make the reading go up or down... not sure if it's a linear or inverse voltage vs corresponding temperature type sensor).
I recommend start by unplugging the sensor at the front of the car, with key on use a DVOM to check the voltage and ground at the sensor's plug. If it's a 2 wire sensor, it will be a 5v and ground. If it's a 3 wire then one is ground, one is 5v, and the other will be the signal back to the PCM. Make sure it has 5v and ground.
You should be able to check the temp sensor reading when the car sits overnight, go out, look at the temperature of the sensor with the key in the ON position but engine not running. It should read same as ambient within 3-5 degrees at most difference.