a router has an UPLINK to another system running the TCP/IP protocol and
several DOWNLINKS to other systems using the same protocol. it also almost
always has NAT services, meaning Network Address Translation
(the UPLINK address is different from all of the DOWNLINK addresses).
a modem (ie Modulaor - demodulator) changes analog telephone into
digital zeros-ones for a SERIAL adaptor connection to ONE system.
Consequently, most routers do not connect to modems and this is not
a technique to share ISP connectivity. When you find a professional grade
router that does have a serial connection, it is used to configure the router
from an internal system also behind a firewall, not using a modem but rather a
straight serial port <--> serial port connection.