The solution is to update the monitor firmware to M2B107. I previously was using M2B105 and M2B106 which both had the same issue. You can obtain the latest firmware on the product download page. I spoke to Dell support and unfortunately they weren't very helpful. At first they told me that the U3421WE is not compatible with a Mac but this is untrue and Dell have a page that explicitly states it is compatibility.
I have a Dell U3421WE monitor connected via the USB-C cable supplied by Dell to my MacBook Pro (using macOS Ventura 13.1). Every now and again the screen goes black/blank and comes back again. I have set the speakers in the U3421WE to be the default output sound device and sometimes I notice when I receive a notification it happens (but I'm not always sure).
U3421WE, MacBook, intermittently goes black