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- 2014 & 2019 CX-5 Touring(s)
You may be referring to a newer model. I'm referring to a 2013 Mazda CX-5 Touring. One press just locks the doors. 2 presses activates the alarm.
Neither my (US-based) 2014 CX-5 Touring, nor my 2019 CX-5 Touring actually HAS a car alarm. Both do the second-press beep thing and both flash a LED on the dash when locked, but Mazda did not include an actual anti-theft alarm with these cars in the US. (I you read the manual, it says "if so equipped".)
But you can test yourself... open a window, lock & 'arm' the car, then reach through the window and open the door from inside.