The routing task (receive packets and forward them) is a layer 3 task.
A router in its basic features is a layer 3 device because it only checks the content at layer 3 (source IP / destination IP).
It doesn't mean that the device is unable to use or check upper layer protocols. For example in a router you can put a filter of TCP ports to block the access to HTTP, FTP, etc.
The router can use a higher layer protocol to send / receive information to/from other routers (RIP, OSPF, etc.)
In addition a router is a host. it can be reached via SSH / telnet or upgraded via FTP /SFTP.