I mean, technically there's no reason a router can't route between more than two networks. For example, I've got both fiber and cable Internet (for no real good reason -- I ought to cancel one and save some money) and I've configured my OpenWRT router to have two different uplinks, reconfiguring one of the four LAN ports to WAN2 instead.

I've also got the other ports configured for separate VLANs (walling my untrustworthy Chinese ONVIF cameras off from being able to phone home, for example), but I think that's technically not "routing" 'cause it's OSI layer 2.

I assume it's not common to have more that two networks being routed, especially in a SOHO environment, but it's definitely not impossible.

Routers can run just fine off only one port too: https://en.wikipedia.org/wiki/Router_on_a_stick

That's not entirely true. Having a router with multiple interfaces is useful for multiple physical networks like VoIP, OOB, dmz, etc. We use vlans a lot for that now, but it's not fair to say only two ports is the only choice.

A router is also an open-by-default device, and knows how to handle routing protocols like OSPF. It's not something you would usually use at home unless you're into that sort of thing. Things we tend to call a "router" are all-in-one firewall/switch/access points.

Only some have an internal switch, others just have multiple NICs.

Are you too used to Cisco devices? Mikrotik routers also have multiple ethernet ports, that are not connected to an internal switch.