RFC 2663 defines[1] a "Basic NAT" as a one-to-one mapping of IP addresses. This can be useful as a way to combine two separate IP networks that share addresses. If you have two networks using the 10.1.x.y range, you could connect them with a Basic NAT so they each see the opposite network as addresses in the 10.2.x.y range.
NAT is just about the Address Translation. Routing is a separate feature, and dropping packets is a separate feature that can be left out, even if those are rare situations.
> but also the actual NAT process
Often you don't even need the NAT process to be involved. Send a packet to a NAT route with the internal destination address, and some routers will simply route the packet directly. Enabling the Source Routing options can also force packets to a specific host by guessing the local address, which is easy when everybody uses 192.168.1.x or other private addresses behind their NAT. Fortunately, this generally isn't possible anymore because LSRR is now usually dropped at the firewall.
If NAT sounds like a mess, it is. NAT is a shameless subversion of the "end-to-end principle, which has set network software development back decades.