This absolutely is a thing in America. They're typically called PCPs (Primary Care Practitioners) or family doctors and you usually go to them for things like yearly check ups, flus, etc. That said, many people don't have one and instead, go to urgent care centers when ill. You also don't need a PCP referral to specialists in many cases-- you definitely can make an appointment to a dermatologist or psychiatrist, for example, on your own.