OK SOLVED.
Bear with me here:
I don't use Apple's Contacts app on my Mac. But for some reason Apple Contacts scooped up all of my contacts, either from Outlook for mac or Google.
This is perhaps my mistake over the years of using too many of these competitive yet overlapping systems on one platform over the years. To be realistic we all know we have 10,000 different type of communication apps on any one of our devices, and all of them want to scrape the other's data, which just so happens to be our personal data... so that ONE App can rule us, until another similar app comes along, scrapes, repeats. It's the world we have built and live in. And constant vigilance is required to make sure our data stays in the app of our choice (and safe), not the app or platform that chooses to scrape our data and take over either without our permission, or because the choices given are so ambiguous that we end up with "Oops, I did it again." (Things in IT were much simpler in Britney's early career.)
In any case, this is where it gets weird: Why would the screen sharing app on Mac Os start putting the names of my contacts in its list...and then not allow them to be removed?
Best I can say with my noob knowledge is "It's a temporary April Fools day feature." OK OK "It's a bug."
In any case, I found the workaround/fix: Delete all contacts from the Apple Contacts app.
And since I don't use Apple Contacts, no problem for me. However I need to be vigilant to be sure Contacts does not try to scrape up my contacts from other locations again (again perhaps my fault in allowing it to do so with all the tinkering I do, aka "Oops I did it again.").
So I am sorry in that I do not have a root cause for this.
But I did find a workaround.
And I am sorry if you have this issue and actually use the Apple Contacts app Because you will still need to figure out why names of your contacts pop up on your Screen Sharing list. Perhaps if it is the name of your Grandmother, a Windows 95 machine will pop up. ;-)
Cheers!
Hans