Is a router a computer or a peripheral of computer?

Linux

Most Definitely A Peripheral. A router in itself does usually contain a microprocessor and some form of memory, but it is still a peripheral.

Modern routers are sophisticated computer devices with storage, memory and network interfaces (possibly including wifi) to perform their routing functions. They may use a custom designed CPU or any of several available processors. Some may have a proprietary software system. Many use the freely available version of Linux or BSD Unix modified to perform routing and firewall filtering functions

Although a router is accessible and configurable from a computer using a network connection (or a serial connection), it is an independent operating device with its own computing capability and is not a peripheral device to any specific computer.