My understanding from the show alone was that you got airbender tattoos only if you were an airbender who had mastered airbending. According to the Avatar wiki, this entailed mastering 36 tiers of airbending OR creating your own technique; Aang mastered 35 and created the air scooter. The source is the Lost Scrolls, one of the short novels or comics which I presume are actually canon. Makes you wonder what the 36th tier is and if Aang ever learned it and passed it down to Tenzin.