It is a coming of age thing but also has religious overtones, since you have to purify your body, meditate on the gods, etc, before undergoing the ritual. The odd thing is that the Dalish claim it distinguishes them from the Shemlen but the old barbarian tribes used tattoos to differentiate between clans and the Chassind still do so, However, the Orlesians no longer use tattoos, nor are they in common use among other humans who follow the Chantry. So may be there is some cultural or possibly even religious reason why the followers of the Chantry no longer have tattoos. This would then suggest that the Dalish adopted tattoos to distinguish those who worshipped the old gods from those who worshipped the Maker. In which case it is not an ancient elven practice but a more recent development.
Whatever the case, if you choose not to have tattoos it is either because your were not considered ready by the Keeper of your clan (whether born into the clan or a city elf refugee) or you have deliberately rejected this tradition - in which case you are likely to be an outcast or at the very least treated as an unruly child who is unwilling to accept the responsibilities of becoming a adult.