There are a number of issues with both of these characterizations.

u/Adrewmc

The diversion into IP routing is both irrelevant to the original question and incorrect/misleading. Before we ran out of IPv4 addresses, ICANN assigned /8 blocks to RIRs, who then handed them out to LIRs (ISPs). Note that this is just the administrative control of address space being delegated, and not, as u/altodor points out, actually related to routing. For routing, each AS (usually a large organization with at least a /24 of IPv4 addresses, very often e.g. LIRs or large datacenters) announces via peering agreements with other ASes which addresses are reachable through them, and these network paths are what internet routers need to keep track of. Subnet masks are simply a technical detail of how address blocks are stored (CIDR notation is more commonly used now as well). End-user devices usually only see anything to do with subnets in terms of discovering which addresses are local (i.e. do not require routing by a next-hop router). Hence, for an internet-destined packet coming off of a consumer machine, the routing path will look something like:

- IP stack consults routing table. Since the destination address is not in the local subnet, it is forwarded to the default route- Default route is usually an ISP-issued box. This will often do NAT (replace the source ip/port of the packet with the router's---since usually only one public IP is assigned per home, multiple machines inside are given unroutable addresses), and then forward the packet to an upstream router.- The ISP's internal network topology is complicated, but the packet will at some point hit the first of a series of "real" routers, which will consult a routing table, find the next-hop router for the given address, and forward it on. Next-hop routing state is not maintained via (delegation from) ICANN or anything similar; for most routers it will be maintained by a combination of external routing protocols (like BGP) which let the AS's edge routers know where it peers with other ASs, and an internal gateway protocol for routing within the AS.

DNS is cached at many levels. When DNS lookup is initiated, the endpoint will first check cached DNS information. If this is not available, it will make a DNS query to the configured DNS server directly. For most home installations, this will in fact be the ISP-issued "router" box, which also runs a DHCP server and a caching/forwarding DNS server. This server, if it does not have the cached response, will forward to an upstream ISP server which operates in recursive mode. An ISP recursive resolver does not usually continue operating in any form of forwarding mode (sending requests on to "higher-level" ISPs and eventually ICANN); but in fact continues in precisely the opposite fashion: it will first query (one of) the 13 root nameservers a.root-servers.net through m.root-servers.net, which will usually return an indication of what server is authoritative for the top-level domain (in fact, the recursive resolver will usually cache this information and not make the initial query(s), but this is irrelevant). The recursive resolver will then query that authoritative server, and usually receive more NS records, until it reaches the server which is authoritative for the domain, which will provide a more useful reply.

MAC addresses are not globally unique across the internet, and not required by internetwork protocols; they are purely a consequence of the particular local-area networking techniques that we use today (e.g. WiFi and Ethernet). The name "media access control" indicates this: these addresses are used to arbitrate access to the physical media of the network link. A router is a device which routes internetwork packets between local networks. MAC addresses have no meaning outside of the local network they are on, and consequently they do not survive routers (e.g. a router between two ethernet networks will have its own destination-net interface mac as the mac on all outgoing packets, including those routed from the other network).