Which Machines can I mount?

When starting the game, I was under the impression that you could mount any overridden Machine, regardless of size (like riding a Behemoth in Shadow of Mordor). However, out of the handful of species of Machines I've overridden, I've only been able to mount the Striders. When I overrode Watchers or Sawtooths (Sawteeth?), I wasn't given an option to Mount them.

Are you able to ride multiple types of Machines or are Striders the only mountable ones?

There are only a handful of mountable Machines. Based on this article, it looks like the only mountable ones are:

• Striders
• Broadheads
• Chargers

None of the others are able to be mounted, but will instead become friendly and attack hostile Machines on sight.

It's worth noting that your mount is not permanent by default, however, there is a perk that enables you to have a permanent mount that you can call at any time if you don't actually have an active mount, as mentioned in this guide:

But there is an easy way to unlock permanent mounts. They can also help you in battle! You can call a mount whenever you want, how often you want. Here’s how to do it:

• In the Skill Menu buy “Call Mount +” (Forager Skill Tree).

To supplement the excellent accepted answer with in-game evidence, that there are only three machines that may be mounted is evidenced by this loading tip:

The text says:

If you're looking for a ride, consult the Map to locate Striders, Broadheads or Chargers. They often graze near settlements.