The router MUST ATLEAST HAVE 2 DIFFERENT PHYSICAL/MAC ADDRESSES, given it connects 2 different networks, having different architecture... ( peer protocols come in use ).
NOW,
talking about router's multiple ip addresses, one for each interface....
One must recall that a router is primarily a network layer device, like the 3-layer switch ( a kind of router ), although it can and is often used at 2nd layer also ( private network routers - termed intradomain routers )
And the TCP/IP suite requires the network layer devices to handle the flow of packets ( since the internet is a packet switched network - packets being called datagram ) across asynchronous systems ( interdomain routing )....
Thus, a router or any third layer device ( specifically the interdomain ones - acting as gateways between 2 asynchronous systems )....needs to be able to correctly & EFFICIENTLY DETERMINE the route ( the next node in its immediate neighborhood network i.e. physically linked )....
The keyword is " efficiently "...
Now, to do that, the routers , specifically at the interdomain level, need to communicate - about their local networks, status of their neighborhood routers ( directly connected ), their own status like heavy traffic , busy/closed interfaces, etc...
ALL OF THIS COMMUNICATION FOR - getting a datagram packet from any corner of the internet to any other corner....
There are dedicated international and peering agreements ( local ISPs ) called protocols for these router-router communication ( inter and intradomain ).
These are the protocols which help every host - end devices , routers etc, connected to the internet to maintain their routing tables..... and which is the software level BACKBONE of the entire internet....
Now, consider, one router R1 in any area A1 goes down for some reason, this will immediately alert its neighboring routers ( thanks to protocols ) and they will send the message to ALL THE OTHER ROUTERS CONNECTED TO THE INTERNET.... let say this router was the only one which connected all the local devices in area A1 to the internet.
THE FINAL AHA MOMENT -
how does the routers which have ENTIRELY DIFFERENT ARCHITECTURE FROM THE R1 router know which particular router ( MORE ACCURATELY - which interfaces/path-ways/routes ) are now unable to accept any packets and hence to be aware in case any packet having destination at A1 arrives, i.e. that packet must be dropped and send a host-unavailable message to the sender....
Since, routers already have a lot to deal with, and being able to process the address version of different architectures is going to be REALLY REALLY inefficient in a internet level WAN ...
SO obviously we need a way to be able to address basically any router being connected with a single set of addressing system... thus we need a logical address... AHA - enter ip addresses for each interface of the router....
Obviously, in case a specific interface goes down or becomes heavily congested.. that particular path need only be communicated to other routers or the routers which use that path..
So , there you have it.... why routers need ip addresses too with physical address... for each interface