If IP addresses are unique, why is it so many routers have the address 192.168.1.1?

An IP address is not necessarily an "identity on the internet", it is just a unique address assigned to a specific network card on an IP network. The internet is one big network, and so anything on it (that has an IP address) has a unique address, and so the IP address can be considered its identity.

Your router is on the internet, and so it has an IP address unique to it.

Anything on the inside of your network is not on the internet, it is on your LAN, an entirely separate network, with its own address space. Anything on your LAN must have a unique (within the LAN) IP address to participate properly with your local network.

Anyone else who has a LAN is also not part of the internet, and not part of your network, and so can use the same addresses you do.

In order for your LAN devices to talk to the internet, their outgoing traffic has the source address changed to match that of the internet IP address of the router. The router keeps track of this, and makes sure any response traffic gets sent to the right internal machine. This is called Network Address Translation.

Imagine that your router at home has the address 192.168.1.1 and your PC has the address 192.168.1.100. Then another router at your friend's house also has the address 192.168.1.1 and his PC has the address 192.168.1.100.

Addresses that begin with 10, 172.16 - 172.31 or 192.168 are referred to as private range addresses (class A, B and C). These addresses are used in a limited area of the Internet, namely within your own local network (LAN) at home which you have control over. They don't need to be registered. Internet service providers usually use different address range, and the addresses they use are registered with an Internet authority.

In the example above your public address might be 109.240.120.13 and your friends public address might be 83.42.112.110. These are the addresses that you and your friend use on the Internet, and they are unique.

RFC 1918 defines 3 ranges of IP addresses that are "private", for use internally within any given organization. These ranges are 10.0.0.0/8, 172.16.0.0/16, and 192.168.0.0/16.

You will note that while many routers use these address ranges for their internal LAN networks, the router WAN interface is assigned a unique publically-routable IP address. when users of the internal network access the internet, they masquerade as the router's public address, using NAT.