Why does a router serial interface don't have a mac address?
Let's take a step backwards and ask why a MAC address is not relevant on a serial interface.
MAC is Media Access Control. A MAC address is relevant in Multi-Access networks like Ethernet where you have several hosts on the same network. In such a network, you have to do an ARP broadcast to find the MAC address of the host, so you can send packets address to it and it only.
A serial line has only two ends, so it is fairly obvious where to send the packets. That is why a serial line doesn't have a MAC address. Usually. Or does it?
If it is a straight point-to-point connection like HDLC, there is no need for a MAC address 'cos you know to send the packets to the guy at the other end of the link.
But consider a multi-access serial link like Frame Relay. Does that have a MAC address? Well yes, it does, sort of ... it is called the DLCI, or channel number. That is why in Frame Relay, you either have to have a static ARP mapping, or rely on frame-relay inverse-arp, or tie the (sub-)interface to a single MAC address with frame-relay interface-dlci.
There are other instance of (theoretically) multi-access serial interfaces, such as SDLC, but that is enough for now.