I think you need to distinguish two separate attributes here: First, there are team sports, meaning multiple individuals work together, vs. individual sports. And second, there are sports where the participants react to each others' actions vs. sports where the participants' performances are each measured on some objective scale and the best performer wins.
As you pointed out, there are many team sports, in my sense, involving more than one team: track relays, swim relays, ski relays, team golf, team gymnastics, etc. You can basically take any individual sport and make a team sport like this out of it. But those are all sports where each performance is measured separately.
But what your question was really aiming at, I think, is that there are apparently very few "mutually reactive" sports involving more than two parties. This applies to both team sports and individual sports. For example, individual sports such as tennis are normally played between two parties as well.
There are some activities where you have multiple individuals all playing against each other, especially if you are willing to stretch the definition of "sport" a bit:
- most card games
- most board games
- some billiard variants
- some leisure table tennis variants
- computer games
- WWE ;-)
For some card, board, and computer games, you can also have more than two teams playing against each other, but here you're getting quite far away from sports.
I would also add
- long distance running
- bicycle racing
and similar sports (e.g., cross-country skiing) because the importance of drag and/or pacemaking (rabbits) causes the participants to react to each others' moves, as opposed to just individually running as fast as they can, as in short distance running. Each of these can be done in teams, not only as relays (more common in running, skiing), but also with all participants going at the same time (as is typical in cycling).
Bicycle racing is a bit of a special case, because only one individual wins. But they have teams, and the team's goal is to have one of their guys win, so it's almost the same thing. Biathlon used to have team races where all four individuals from a team went at the same time, but the teams did not all start at the same time but were timed separately. Speed skating has a team race, but the teams are at the opposite ends of the track, so they do not interact. Short-track speed skating has relays, so the team does not cooperate within itself. There might not be a crystal-clear example of what you are looking for. But you could plausibly organize one, like a street race (marathon, say) for teams.
Looking at those lists and thinking about it, I think the reasons why there aren't many clear examples of sports involving more than two teams playing against each other are mainly practical. First, there are reasons of geometry. Normally, opposing players and teams will face each other. If you have more than two parties, you are very limited in what you can do. Second, there are reasons of safety. Most sports in this category involve balls, because that's how players play off each other, and if you have balls flying at high speeds in all directions, it could become dangerous. Also, player contact will be dangerous if it is likely that more than one player can touch, tackle, or hit you.
The best way to do this, then, is with sports that are slow, turn-based, and work well with a flexible or circular player arrangement. The racing examples fit this as well, because the movement of the participants relative to each other is usually slow, safe, and flexible. Or you escape from geometry and safety constraints by using computers.