iPad Pro (3rd gen) as a 2nd monitor for MacBook Pro Monterey not working?

Okay, as promised, I'm home now. When following the steps for Monterey in the article that I linked for you, when you open Displays in System Preferences, just under my MacBook Air display in the left side of the dialog box, there is a dropdown for Add Display, and you can see both my Samsung TV and my iPad Pro 11 listed, as seen in this image.

After selecting the iPad Pro 11, the setting changes to this,

and I can use the Display Settings to determine what I want to do with the two displays, the main display and the iPad. I believe you may be looking for an actual setting named Sidecar and that does not seem to exist any longer. I believe that may have changed with the implementation of Universal Control.

I hope this helps you. If this does not help, use the Get Support link at the top of the page here to contact Apple Support. If you have an Apple Store close by, use this link to make a Genius Bar appointment to have the hardware examined by Apple. They will explain your options. Genius Bar Reservation and Apple Support Options - Apple

A little additional info: if you search for "Sidecar" in Preferences, you won't find it -- at least not by name.

It's enabled automatically(if you've enabled all the Continuity requirements). Searching for "Universal Control" in preferences will show you the settings to turn on to enable "Universal Control" (which needs most the Continuity requirements as well) ... so you may find it possible to choose which feature you want to use.

To see which is available (Sidecar or Universal Control), look at Preferences -> Displays then the "Add Display" dropdown. If you see something like this:

... what it means is, in this case either Universal Control or Sidecar is available (even though you can only use one of those at a time). Which one you choose depends upon what you want:

Either:

To use both the Mac and iPad as separate systems, each with their own apps, sharing a single keyboard.

To use just the Mac, with the iPad's screen acting as a dumb external display for the Mac.

If you select "Link Keyboard and Mouse" with an iPad, you'll get Universal Control. Alternatively, simply try to drag your cursor off one side or other of your main display. Either way, an image of your iPad's screen should show up next to your main display.

With Universal Control, your iPad continues to run your iPad apps as a separate system, and all the gestures continue to work. But if and when you want to use a keyboard & mouse on the iPad, just move the cursor from your Mac over to the iPad, and the Mac's keyboard & mouse becomes the iPad's keyboard & mouse.

If, instead, you select "Mirror or extend to" an iPad then it becomes Sidecar, which essentially just turns it into an external display for your Mac. Whatever iPad apps might be running are suspended, and you'll only see whatever Mac apps you're running on your Mac mirrored or "extended" from the Mac's main screen.

(In the picture, the other choice is an external monitor which happens to be hooked up to an Apple TV named "Home Office". If you select that, it effectively dumbs down the Apple TV and connects through to its external display. As if you unplugged the external display from the Apple TV and pluged it directly into the Mac.)

If you only see one (or no) choice for your iPad, then your systems don't meet each and every requirement for one or both Universal Control and/or Sidecar.

[Note that this information and the Preference settings will likely change in future versions past MacOS 12.3.1 and iPadOS 15.4.1.]

Just to complete the story, when Universal Control is operating, Preferences -> Displays should look something like this:

If you click on Mirror or extend to .... iPad here, it will turn Universal Control off and add the iPad as a Sidecar instead. After which, you'll probably have to sleep/wake the iPad to quit Sidecar and again be offered a choice of Linking (Universal Control) or Mirror/Extending (Sidecar).

I'm sorry, I'm at work right now, and I don't have my iPad with me so I cannot check. However, I just used it the other day, so I know that it works. Looking at my MacBook Air with Monterey, I do not see the settings to turn on Sidecar when I don't have it. I followed the steps in that support document and it worked just fine. I'm guessing you have both devices updated to the latest version of iPadOS and Monterey? Have you tried restarting both devices?

I'll be home from work in about 4 hours and will attempt to connect my devices and will let you know if I encounter a problem or provide steps on how to connect them.

Thank you for taking the time to send me that information, however it is very similar to what I posted, and I am not experiencing any difficulty with Sidecar or Universal Control. I was responding to the OP who was experiencing the problem. Hopefully, between what you and I posted, he understands and is able to use his devices.

See this information here, Use an iPad as a second display for a Mac - Apple Support