Confirmed that popping up a new browser window on a second display CANNOT be done anymore.
When I was researching at the beginning. Many JavaScript forums had codes shown to do this and detailed how to detect the second monitor, which dated mostly from 2012 to 2018. So I thought this feature was doable and with so many years passed it would only get easier to implement.
However, when I tried to detect the width and height of a second display it could never find it no matter how I tweaked the codes. I thought JavaScript function names might have changed or it became browser-dependent. So I kept on searching and experimenting but none worked.
Eventually, I tried instead to push the pop-up browser window out of the current display to left and right, without regards of the second display's resolution, then I found out that the pop-up windows always stayed on the left or right edge of the current screen no matter how far off you tried to push it. This well-formed feature gave me a hint that this limitation (not passing the boundaries of the current display) might just be a built-in feature within the browser on Windows platform.
Knowing this fact, further searches in a different direction brought me other users' experiences in more recent years and confirmed that those JavaScript functions detecting a second display would not work anymore on Windows 10 platform at least dating back in 2019 on Windows 10 platform.
Maybe this restriction is now considered a security necessity because a browser is largely dealing with unknown users from the Internet and it should naturally be limited on what it is allowed to detect and access on the computer’s properties. It's now the end of 2022, so I might just conclude that this kind of pop-up on second display implementation is a dead-end.
(I know there are already specialized sign-off devices on the cashier's counter top in large department stores but those are essentially provided by payment system providers, which require purchase and installation of separated software/hardware/devices. This would not be desired as we have already had a web application that is fully handling the entire ordering processes from start to finish. Our sign-off feature is just a new requirement.)
I considered Somrlik's solution using AJAX but it requires a call to check every 5 seconds or so which is quite a work load for a process happens only 5 to 10 time during the whole day of opening hours (not all sales require a signature). Moreover, it also requires a dummy web page opening all the time to stand by on the customer facing display which should be blank most of the time. Therefore, this solution may not apply in my situation.
I wish I were wrong with all this and someone could point me a way to implement this because I do have a need to pop up a separated web page on a second display facing customers for them to sign off an order when a sales person clicked a "Sign Off" button on the ordering page on her sales computer screen.