The answer to your question is no, they are not the same. AISI316 essentially has no meaning in current times because AISI no longer maintains standards, and even when they did, it was for composition only. ASTM A213 is a product standard that covers composition, heat treatment, mechanical properties, and other characteristics. ASTM A213 does include 5 flavors of 316, so it looks like you have identified a standard that could work.