Solution:
I had the same issue - the pop up appeared to markup the snip if I captured it using the "New" button, but not if I started the snipping tool with the shortcut keys Windows + Left Shift + S.
I am using an external display, duplicating the laptop display to an external monitor.
I noticed in my Notifications panel that Do Not Disturb was turned on, but I didn't turn it on and didn't want it on.
I changed this setting and it resolved the issue:
Settings > System > Notifications > Turn on Do Not Disturb Automatically > When duplicating your display (priority notification banners are also hidden) > UNCHECK THE BOX
Now when I take a screenshot with Windows + Left Shift + S a pop-up appears at the bottom right and I can use the button "Markup and Share" to edit the screen shot.