Yes, there are different and parallel institutions in which a specific solicitude doesn’t affect the other. In fact, the “foreign investor ID” is granted by the Chilean tax department (Servicio de Impuestos Internos) while permits are given by “Foreign and Immigration department”.
They also differ in how fast they can be obtained. You can apply and obtain the “foreign investor ID” immediately, yet your application for a Visa or permit can take somewhere between 6 and 10 months.