My starter pet just became 18 days old and they did not become an adult. Is the age mechanic different for starter pets than they are with other pets?

Maturity is determined by the maturity value, which unfortunately is kind of difficult to find or in certain situations, completely masked.

For generated pets (Starters included), you can actually look at its full image URL to find this value.

https://www.sylestia.com/image_builder_pet.php?species=draeyl&breed=sylesti&gender=female&gd1=NN&gd2=NN&gd3=NN&md1=AA&md2=GG&md3=NN&ec=4e1542&c1=e71724&c2=de5069&c3=d7e410&gd1c=dc659a&gd2c=85a143&gd3c=8d3688&md1c=91af42&md2c=aa9f2d&md3c=c3dd68&ac=34e45e&ac2=66031e&maturity=646&costume=0&size=250&owned=1


A pet becomes an adult at a maturity value of 648, which represents hours. 648 / 24 = 27 days.

So your pet will become an adult in less than 2 hours. =)