I know walnut oil should be stored in the fridge. Is the same also true for peanut oil?
No. Peanut oil has a similar fat profile to that of sunflower, corn, and soybean oils, all of which do not require refrigeration after opening and are typically used for frying. Walnut and other edible nut-based oils more easily oxidize after exposure to air, light, or high temperatures. They are primarily used for their light, subtle flavors. Any oil will go rancid given enough time so unless you're using your peanut oil very infrequently, there is no need to store it in the refrigerator.
There is not need to store peanut oil in the fridge. If still in its original container, it will last for up to 2 years in a cool dark place, like the bottom shelf of your pantry. Once used, you can reuse, with a recommended discard date of 6 months after first use.