Is it possible to mirror only one space or one desktop to a second monitor/screen?

It is not a problem as macOS has never supported this functionality, macOS cannot mirror a Space to a physical monitor. You can submit Apple product feedback and make the case for a feature request. No guarantee Apple will eventually implement it but you never know. Product Feedback - Apple

Workaround: If you have 3 monitors (laptop + monitor 1 + monitor 2) you can mirror monitor 1 to monitor 2 by going to the Arrangement view on Displays and holding Option key while dragging monitor 1 over monitor 2. That will mirror only those two monitors keeping laptop screen untouched. What you see on monitor 1 is reflected (mirrored) on monitor 2. You can also enable Spaces to be unique across multiple monitors. Settings -> Desktop & Dock -> Displays have separate Spaces.

The original author mentioned facing monitor 2 towards a person sitting across from him as a sort of presentation monitor. Other possibilities might be another room entirely.

I suggested adding a 3rd monitor either with an iPad or a portable compact USB travel monitor which is about the size of an iPad or a bit larger (14"+) as the second monitor for the person controlling the monitors as an observation monitor for what the other person(s) will see on the external monitor facing them. Then mirroring monitor 2 & 3 via the Arrangement Option Drag little known feature to mirror those two displays.

You are going to need a 3rd monitor then go to Settings -> Displays -> Arrange and layout the two monitors and the 3rd. Hold the Option key and drag the 2nd monitor to the 3rd monitor to get only those two mirrored.

You may very well be able to use an iPad as the 2nd monitor mirrored to the 3rd monitor for your audience.

I doubt Apple provides access to the necessary API's for a 3rd Party to enable what you wish to accomplish.

Besides using an iPad, there are quite a few USB-C based super thin portable monitors on the market. Designed to be portable second displays for a laptop on the go. These take up a very small amount of space. Dell, HP, Lenovo, etc. all offer these products. They are also much cheaper than an iPad.

Another option is to use Airplay over an Apple TV. You could treat the TV as an extra monitor and it would be wireless. But you would still need some other 2nd display to mirror with it. Or you use a TV that can act as a monitor and you run a USB-C to HDMI cable to it. If the distance is too great there are solutions to extend HDMI.

Yet another option just occurred to me. Some ultra-wide screen monitors have a feature to split the screen between two video inputs. Placing one input on one side and the other input on the other.

So thinking you connect the Mac twice to such a monitor. USB-DisplayPort cable and perhaps a USB-C Thunderbolt cable. Other options would be 2 DP cables or a DP and an HDMI. That would give you two monitors in one so to speak. The Mac won't know nor care that both inputs are on the same monitor. Each connection is on a different input to the monitor. You can toggle the full screen inputs or use the split screen as needed using the monitor controls or possibly software from the hardware vendor such as Dell who offers a Mac version of their Display Manager.

yes, we have the same problem