@Switch Blayde
That's how I discovered the ethernet cable from my MacBook went to the router. I had thought it went directly to the cable modem.
I depends on the configuration of the router and the modem.
Internet service modems come in different varieties and have different configurations. Your modem could be only a modem (modulator/demodulator) and be a pass-thru configuration where it doesn't do anything other than convert the network signals into IP packets and then you would need a router configured with your username and possibly a password to give you the service you paid for.
Most of the time though these days, the modem itself has a router built in (or more accurately a router with a built-in modem). That router/modem would contain the account configuration and has a built-in DHCP server to provide network configuration to connected devices.
If your internet provider gave you a modem AND a router, then the modem is pass-thru and the router is required for the authentication and network communications with your Mac or any other devices that you may connect to the router.
If you got a router on your own, separately from your internet provider and you plugged it in without any specific configuration, then you could connect your Mac directly to the modem.
Although, if you really want to, you could connect your Mac to a pass-thru modem, but you'll have to configure the Mac with the PPP software required by your internet provider depending on their protocols. Most users can't do this and would require a technician from the internet company to come and configure your Mac to connect to the internet.
When I was living in Canada, I had both types of services at my house. A fiber optic modem/router/wifi combo for the family from Bell. It was connected to the fiberoptic cable and the electrical outlet and everybody connected by wifi. It had 4 ethernet ports that could be used. I also had a second internet line with a pass-thru modem for a service with a static IP address. I had my own router that I purchased which had enterprise level services like private VPN, and that was connected via ethernet to my desktop Mac. Let me tell you, it was a bitch to configure that router to get optimum speed, down to the packet/chunk size.