Why does car interior stay cold while car is "warming up"

Hello, the heater core will only have coolant flow through it at 200 degrees or so when the engine is at full operating temperature. This could indeed take about 5-10 minutes of driving. I would suggest, to gain more comfort and clarity, that you have your vehicle checked by a local Honda dealership, as factory warranty is likely still in effect, and the dealership can address your concern first hand.