The Linux box presumably uses the resolver as announced by the DHCP server on the router, which I would guess is 8.8.8.8. Therefore, the chain goes linux-box → 8.8.8.8 → auth servers, 8.8.8.8 being the resolver.
Or perhaps the router announces itself as the resolver, and proxies all requests to 8.8.8.8. This would not be necessary, but I guess it would be useful for the router to do proxying instead of NAT whenever the protocol (DNS in this case) makes it possible. There is still no reason to call the router a “resolver”, or even assume that the DNS proxy software on the router even has a cache.
Describing DNS as a stack of resolvers is like describing HTTP as always having a chain of caching proxies. It is not normally true, and even when it is close to true, it’s made to not look that way – i.e. it’s not useful to think of it that way, especially when explaining it to someone else.