I've run into the same problem... The 2 "work arounds" I've found so far when the "Copy as Document Link" pastes just an icon:
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Not the desired / best way -- Save the document I'm working on a draft (if email) and restart Notes. Go back into my draft and paste the next link
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Go into any document and simply right-click / copy. Now the next time you use the "Copy as Document Link" it should function as normal.
It's almost as if you need to clear the clipboard first before you can copy/paste a 2nd, 3rd, etc document link... Weird
I would love to hear if anyone finds a permanent fix for this though. Definitely an annoying quirk.