No. A point-to-point link does not have to have its own IP address. Point-to-point links can be unnumbered.
You can, for example, have one router with an IP address of 192.168.120.3 on the 192.168.120.0/24 LAN and another router with an IP address of 192.168.121.9 on the 192.168.121.0/24 LAN. They can have a serial link with one end numbered 192.168.120.3 and the other 192.168.121.9 without needing additional IP addresses just for the link.
Logically, to enable IP on interface S0, you would need to configure a unique IP address on it. However, it is also possible to enable IP on the Serial interface and bring it up without assigning a unique IP address to it. This is done by borrowing an IP address already configured on one of the router's other interfaces. To do this, the ip unnumbered interface mode command is used as shown below. -- Understanding and Configuring the ip unnumbered Command