Combine chicken breasts and broth in a multi-functional pressure cooker (such as Instant Pot). Close and lock the lid. Select high pressure according to manufacturer's instructions; set the timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions for 10 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Transfer chicken to a bowl and shred.
Select Sauté function. Add cream cheese, Sriracha, ranch dressing, and chile-garlic sauce to the pot. Stir in shredded chicken. Add Cheddar cheese a little at a time, constantly stirring until melted and all ingredients are well combined. Select Keep Warm function until ready to serve. Garnish with green onions.