Well first off, if you're going by Persona 5 rules, you can't own both a Persona and a Shadow.
Secondly, your shadow isn't named anything it's just your inner darkness made manifest, say "Shadow Idon" or something, not given an actual name like Arsene.
Third, personas within the games are often based around some sort of mythology, Greek (twice), Japanese, Picaresque. For Joker(DC) specifically, a really cool persona that could parallel his insanity could be someone like the Mad Hatter from Alice in Wonderland.