For Lorelai, eating the fancy burgers and wearing the party hats isn't just about fun; it’s a sign that you’ve been "assimilated" into the rigid world of Emily and Richard Gilmore.
“You’re a Gilmore. It’s in your blood. You can’t escape it, and I can’t protect you from it anymore.” [S1E17] Ate The Hamburgers, Wearing The Hats
The phrase is Lorelai’s shorthand for selling out. For Lorelai, eating the fancy burgers and wearing
Lorelai is terrified that Rory is becoming too comfortable with the "high society" lifestyle she herself ran away from. Seeing Rory fit in seamlessly with Richard’s business associates triggers Lorelai’s deepest insecurities. performative rituals of the wealthy.
It refers to the idea of participating in the silly, performative rituals of the wealthy.