Trivia: Strictly speaking a box that does NAT is not a router in the IP protocol sense, it's a kind of proxy. The router requirements RFC explicitly forbids altering most fields (incl the address field) in the IP header.
The NAT RFCs came after the routing RFC and refer to NAT as a router function not as an orthogonal function, boxes that do NAT are referred to as routers in the RFC. This is reflected in the real world where NAT is implemented as part of the routing chain not as a separate module. Remember NAT isn't a box creating 2 sockets and ferrying data between them it is just the translation of fields on top of normal routing f…
This is not the case, certainly if you are referring to the router requirements. Last I looked, the rest of the IETF was also very cognizant of the distinction, as there has been wide anti-NAT sentiment in the IETF, trying to get people to move over to IPv6 etc.