Given that the Dalish don't become priests of their gods once they get the tattoos, I don't think they devote themselves to each and every attribute of that god.
We tend to overlook the importance of the different sides of a given deity, and think that only one aspect defines them, something that wasn't true for ancient polytheistic societies. Let's take ancient Greece. Their gods had different names and epithets, sometimes because of the place of their worship or because of their actions or roles. For example, depending on the place, offering or moment, Zeus was revered as Olympios (in his role of king of the gods), or Horkios (keeper of oaths), or Xenios (patron of hospitality), etc.
Although the elven pantheon is obviously far less developed than the Olympian one, we do know that the Evanuris have different attributes:
-Elgar'nan: vengeance, sun, All-Father.
-Mythal: protector, love, All-Mother.
-Falon'Din: death, fortune.
-Dirthamen: secrets, knowledge.
-Andruil: hunt, sacrifice, the Vir Tanadahl.
-Sylaise: domestic arts, the Hearhtkeeper, the Vir Atish'an.
-June: Master of Crafts.
-Ghilan'nain: navigation, Mother of the Halla.
It could be that, the same as ancient temples in Greece were dedicated to a specific attribute of the gods (most famously, the Parthenon), young Dalish may dedicate their tattoos to a specific aspect of their chosen god. A warrior who may not be the best archer in the clan could devote himself or herself to the Way of Three Trees and choose Andruil. A hunter who wants to never get lost in the forest may choose Ghilan'nain. A healer that loves study may choose Dirthamen. Etc.