How to use Target Display Mode with late 2013 display iMac with 2018 Macbook Pro

Did you read this?

iMac models introduced in 2011, 2012, 2013, and mid 2014

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced
• in 2019 or earlier and have macOS Catalina or earlier installed.
• The cable connecting the two Mac computers must be a Thunderbolt  or Thunderbolt 2  cable.

Just purchase a lower-price point 32-inch display offering an HDMI port and purchase a USB-C to HDMI 2.x cable, or If you can find a lower-cost 4K UHD display with HDMI, then Belkin makes a USB-C to HDMI 4K HDR adapter and use the aforementioned HDMI cable.

+1 on Jack-19's accurate advice. If the MBP needs an external display, please buy one you will be much happier. That solution is less expensive (does not require expensive cabling), is dead reliable, and is simple to setup. Target Display Mode is no longer supported and has not been for years now.

Are you sure you're pressing Cmd+F2 on iMac? I have 2019 MBP with Big Sur and old 2013 iMac with Catalina and TDM work fine with this pair (with a caveat). I actually find this post researching if I'm gonna loose TDM if I upgrade to Monterey.

Even though they say your other Mac must have Catalina or earlier, that's not entirely true. I use TDM with Big Sur on my 2019 MacBook Pro all the time. I guess it's just "not supported" anymore. There is a caveat though: quite often after pressing Cmd+F2 the iMac does not turn it's backlight. Turning TMD on and off several times helps. Also I didn't manage to get the sound working through TDM (I think it wasn't working on Catalina either).

Welcome to the Apple Support Communities.

Sorry, no. macOS Monterey does not support Target Display Mode.

Jack