It's the mayo.
Ketchup or special sauce are standard. Mustard is Texas style.
This is kind of outdated, but in the 80s mayo on a burger was weird.
A lot of places California burger means avocado, too.
It must be a regional thing to my area then. You were the last reply and the consensus is that's just a regular burger or dressed burger and avocado would make California. I agree the mayo is the only separate thing about it. I know of at least 4 burger pizza places that call it California and have it like I described. One place puts it in a hoagie roll and calls it a gizmo.
Oh yeah. The "California" sandwich of the 80s was (alfalfa) sprouts, turkey, and avo.