Classical circuit switching means a separate wire for each connection. This is still in common use for residential telephone connections: each subscriber has a dedicated wire to the Central Office. But a separate physical line for each connection is not a solution that scales well.
Once upon a time it was not uncommon to link computers with serial lines, rather than packet networks. This was most often done for file transfers, but telnet logins were also done this way. The problem with this approach is that the line had to be dedicated to one application (or one user) at a time.
Packet switching naturally implements multiplexing (sharing) on links; the demultiplexer is the destination address. Port numbers allow demultiplexing of multiple streams to same destination host.
There are other ways for multiple channels to share a single wire. One approach is frequency-division multiplexing, or putting each channel on a different carrier frequency. Analog cable television did this. Some fiber-optic protocols also do this, calling it wavelength-division multiplexing.
But perhaps the most pervasive alternative to packets is the voice telephone system’s time division multiplexing, or TDM, sometimes prefixed with the adjective synchronous. The idea is that we decide on a number of channels, N, and the length of a timeslice, T, and allow each sender to send over the channel for time T, with the senders taking turns in round-robin style. Each sender gets to send for time T at regular intervals of NT, thus receiving 1/N of the total bandwidth. The timeslices consume no bandwidth on headers or addresses, although sometimes there is a small amount of space dedicated to maintaining synchronization between the two endpoints. Here is a diagram of sending with N=8:
Note, however, that if a sender has nothing to send, its timeslice cannot be used by another sender. Because so much data traffic is bursty, involving considerable idle periods, TDM has traditionally been rejected for data networks.