There are a few reasons why Archie might want to seem like he's connected to the Southside Serpents. Maybe Jughead wants to finally blend his two worlds together, and having Archie as an honorary Serpent will make it easier for Archie to stay safe in snakeland. But the more dire reason is that Archie is headed to prison for a murder he did not commit, as it was teased in the season 2 finale, and that he'll need to seem tougher and "affiliated" in order to survive behind bars.