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Since I have never worked with a dial-up modem, I was wondering if it created an internet connection for only one user, a point-to-point connection, or if it also supported multiple users.

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    A modem connected to a router can support multiple users -- albeit glacially slow. Modems are most commonly connected to the end-user device -- laptop, ATM, credit card terminal, ... -- thus "1 user". – Ricky Beam Jan 26 '18 at 21:55
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Almost all consumer Internet access links use only a single IP address and many business links as well. A single IP address can't be shared directly but extremely commonly (source) NAT is used.

A modem per se can't be shared but when a connection is dialed up by a router and then NATted, the connection can be shared just with xDSL, fiber or anything else. From the network perspective*, the modem is just the line interface like a fiber transceiver or a cable modem, not much else. It's up the device using it what you can do with it.

Actually using single-line analog and ISDN dial-up connections to share to a whole network wasn't uncommon until the late 1990s. Hard to imagine nowadays...

*"from the network perspective" = from the packet network perspective - A modem provides a serial link to an overlay network on top of the telephone network. The PPP protocol on top of the serial link converts the serial line to a multiplexed 1:N (or N:N) packet network interface and running IP on top of that eventually connects to the Internet.

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In the 1980s, most of the dial-up connections I saw were UUCP, not IP. Our server would call the various other servers and exchange mail and files. In this way it most certainly was for more than one user.

By the early 1990s, I saw a lot of corporate networks where there was a dial-up modem making a conventional IP link, with mostly sendmail SMTP traffic and some occasional FTP. This was before HTTP. So that was also multiple users.

By the late 1990s, dial-up was mostly for individuals, and offices had ISDN and leased lines.

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The computer connected to the modem would start one PPP (or earlier, SLIP) session, and it would get one IP address from the ISP. So yes, one connection for every dial-up connection.

There were methods to "share" this with other computers on the same LAN (similar to what we do with Hotspots today).

Of course the speeds would be so low that more than one user could not have done much anyway - and also the need for more than one computer in the home (or for that matter, in the office) with an internet connection was simply not felt :-)

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