I am successfully using Target Display Mode on a 2013 iMac running Catalina.
The source Mac is not running Monterey, however, and I can't help you there but it does work with an earlier OS on the source Mac and Catalina on the Target Display Mac as I shall explain below.
[FWIW I see no reason why Monterey shouldn't drive a Mac in TDM, since it's just functioning as "another display" about which the source Mac
ought to be agnostic. But these are Macs, of course... Maybe introducing a dock in between would help. It's what I have in my setup successfully using a Catalina iMac in TDM.]
Digression over. Yes, Catalina does support Target Display Mode
Until recently there's been one caveat: the Catalina Mac has to be at the login screen in order to function under TDM. This can be from boot/restart, by logging out an active user, or by using Fast User Switching which leaves the user account logged in but inactive. So the Catalina Mac couldn't actually do anything while being used in TDM, but it works perfectly as a monitor.
And, like joevt, above, I found that pressing Cmd+F2
when logged in to a user account displayed a black screen on the Catalina Mac.
It's worth noting that the source Mac detected this as a connected screen, as seen by a change in desktop layout and the presence of a second monitor in the Displays Pref Pane, despite the TDM screen being completely black and immune to brightness adjustments. I have a hypothesis about this, explained further down. Oh yeah, and it similarly took at least two sets of key-presses to switch modes, even with the black screen.
But all that changed yesterday. Now my Catalina iMac happily functions in TDM even when logged in. Music will continue to play, and the keyboard volume controls etc still work - just like the old days when TDM was new and shiny!
I do not know for certain how this change came about. The only clue I have is that I upgraded the Catalina iMac to Safari 15.6.1 just beforehand, and did a full restart as things had gone a bit a bit sucky. Maybe the security content of Safari 15.6.1 also reset an issue with the EFI or SMC on the Catalina iMac? I dunno, but the completely black screen suggests that screen illumination was off but that both Macs still
thought TDM was working correctly.
The source Mac is a 2015 MacBook Air running High Sierra (don't judge me, I like it that way). Here's the full spec of my setup successfully running Target Display Mode on a Mac logged into a Catalina user account:
Source machine: MacBook Air 7,2 (13- inch Early 2015), 2.2GHz i7
Source graphics card: Intel HD 6000 shared graphics (1536 MB)
Source OS: 10.13.6 macOS High Sierra
EFI Firmware: 195.0.0.0.0
Security Updates: XProtect 2162 / Gatekeeper 181 / MRT 1.93
Connection 1: Thunderbolt 2 cable from source Mac to Thunderbolt dock
Connection 2: Thunderbolt 2 cable from Thunderbolt dock to Target Display Mac
In between: Both Macs linked via Elgato Thunderbolt 2 Dock
Target Display Mac: iMac 14,2 (27-inch, Late 2013), 3.5 GHz i7
Target Display graphics card: NVIDIA GeForce GTX 775M 2 GB
Target Display OS: 10.15.7 macOS Catalina
EFI Firmware: 433.140.2.0.0
Security Updates: XProtect 2162 / XProtectRemediator 71 / Gatekeeper 181 / MRT 1.93
NB XProtect and XProtectRemediator were both updated yesterday (22-19-08) along with Safari
To check your EFI and security update status, I recommend downloading SilentKnight and/or SysHist, both highly rated freeware from the excellent
Eclectic Light Company.