It's a real thing.
Deli ears refers to someone who always listens to conversations in which he/she wasn't invited.
Many academics believe it's etymology is rooted in
people who sit in deli restaurants all day and listen to
people around them speak; however, this isn't true. It's beginning, as with all evil things, can be traced back to Oxford University.
After the British subjugated India, they set up a provincial
government in New Delhi. They encouraged natives to bring back any word of rebellion.
One such native, called Tom, was instrumental in stopping uprisings against the British authority. For his deeds, Oxford named the ability to listen to other conversation "Deli Ears," because Tom had ears that looked
like sliced turkey.
The term all but died out until a small cult in Long Island revived it.