Why assign IP addresses to router/switch interfaces?

Question

I get why I would ever want to assign a IP address to a router or switch, for remote login and IP for hosts to reach it. But why assign IP addresses to the interfaces? Is it so the router/switch knows which port to send the packet out? Route summation? But I thought they do that through the routing table, like " that address is out this port".
So why would we ever need to assign IP addresses to specific port interfaces?

Answer 1

Disclaimer
The Author of this posting offers the information contained within this posting without consideration and with the reader's understanding that there's no implied or expressed suitability or fitness for any purpose. Information provided is for informational purposes only and should not be construed as rendering professional advice of any kind. Usage of this posting's information is solely at reader's own risk.
Liability Disclaimer
In no event shall Author be liable for any damages whatsoever (including, without limitation, damages for loss of use, data or profit) arising out of the use or inability to use the posting's information even if Author has been advised of the possibility of such damage.
Posting
You normally assign IP addresses to L3 interfaces so other L3 devices have an IP address to forward traffic to.  (L2 IP address are generally only used for management.)
Suppose you had Host (192.168.1.5/24) <> R1 <> R2 <> (192.168.2.8/24) Host, and you want the two hosts to intercommunicate.  How would you get this to work?
You might started by providing interface IPs on the router interfaces facing the host, such as:
Host (192.168.1.5/24) <> (192.168.1.1/24) R1 <> R2 (192.168.2.1/24) <> (192.168.2.8/24) Host
You then configure "gateway" IPs on both hosts:
Host (192.168.1.5/24 - GW 192.168.1.1) <> (192.168.1.1/24) R1 <> R2 (192.168.2.1/24) <> (192.168.2.8/24 - GW 192.168.2.1) Host
Now each hosts "knows" to send all its off local subnet, traffic physically to the GW IP.  So, for example, if 192.168.1.5 want to sent to 192.168.2.8, it would forward the traffic to the GW IP, 192.168.1.1.  This is a example of why you want an IP on the router's L3 interface.
Next we want R1 to forward the packet to R2, but it too needs a "next hop" IP address, so we assign addresses on the link between the two router, e.g.:
Host (192.168.1.5/24 - GW 192.168.1.1) <> (192.168.1.1/24) R1 (192.168.3.1/24) <> (192.168.3.2/24) R2 (192.168.2.1/24) <> (192.168.2.8/24 - GW 192.168.2.1) Host
R1 then needs to "know" where to send packets with an destination IP network of 192.168.2.0/24, in this case, it need to "know" to send the to IP 192.168.3.2.  When it does, R2, having and interface with 192.168.2.1, will also know 192.168.2.8 can be reached by sending the packet out that interface.
Hopefully, the above will show why IP addresses on router L3 interfaces are needed.
BTW, normally for the R1<>R2 link, you would assign a /30 or /31 network or you might use "unnumbered" interfaces (which "borrow" IPs from another interface).