See my footer.
Keeping it real simple....
issues ISO certificates. Not all registrars can issue the same types of ISO certificates
A Notified Body
issues EC certificates. Not all Notified Bodies can issue the same types of EC certificates, and even if they can, there are often government-mandated limitations as to what the content can be (i.e. by product technology in the case of medical devices)
Most, perhaps all, Notified Bodies are also Registrars, but there isn't a requirement for that. Most Registrars are not, however, also Notified Bodies.
Now both Registrars and Notified Bodies are examples of third-party certification organizations. There are many others. Almost all are commercial operations, unlike say, the FDA.
That said, there is a third-party certification scheme controlled by FDA, so SGS can indeed do some work on behalf of FDA, at the request/option of a manufacturer. We would be paid by the manufacturer for our efforts.
There is also a third-party scheme controlled by the MHLW/PDMA in Japan, so again SGS offers services for that.
And so on. I hope that makes sense.