Does the issue also happen if you use these specific steps to mirror your iPhone screen?
I have no smart TV or other AirPlay capable display.
I have now discovered screen mirroring is not needed. I get the 9:41 AM just by using the iPhone as audio input.
To be clear, this is just plugging the iPhone into the Mac with the same USB-C to Lightning cable that I use to charge or to backup in iTunes. No interaction with the iPhone is needed, I am not using the iPhone's Settings or Control Center or an app or anything else. I use QuickTime Player "File>New Audio Recording", and choose iPhone as the "microphone" input.
Therefore I should not have included "screen mirroring" in the thread title. My aim is only to play music from the iPhone onto my speakers without the need for a Lightning to Headphone adapter.
The problem is only while the QuickTime Player app has a black recording window open; and I do not need to actually start recording, it is enough to just pick the "Start Recording" command, select the iPhone as the "microphone" input, and do nothing else, as if I were just adjusting audio levels. Once closing that window, or leaving it open but selecting a different microphone input, the iPhone's status bar instantly goes back to normal. I can repeatedly switch audio inputs by touching "MacBook Pro Microphone" or "Robert's iPhone SE" using the Mac's touch bar, and the phone's status bar is wrong only when it is selected.
Oddly, the word "Verizon" (my wireless carrier) disappears from the status bar whenever the time is incorrect, and the signal strength changes to 4 bars which is not accurate. The battery icon is no longer green with a "charging" symbol (lighting bolt), instead it changes to a plain black battery, though the percentage is accurate (I watched it go up).
Which iOS version is your iPhone running?
14.4.1 (I updated yesterday)
Which macOS version is your Mac using?
10.13.6
Does your iPhone ever show an incorrect time when the screen isn't being mirrored?
Yes, when using iPhone as an audio input only (see first answer). The earlier reports ( discussions.apple.com/thread/250008253 , /thread/251865467 , and /thread/8088765 ) do not specifically mention mirroring and I shouldn't have put screen mirroring in the title.