There are a total of 18 Daedric artifacts in the game. However, the Skeleton Key does not count for this achievement, and two of the Daedric quests make you choose between two artifacts (although in one of the quests you can exploit a bug to get both), effectively nailing the number down to 15 (or 16).
You don't need to possess all 15 artifacts at the same time; as long as you have picked up 15 artifacts in your playthrough, it should count.
https://elderscrolls.fandom.com/wiki/Daedric_Artifacts_(Skyr...)