I've seen this lots of times. It's your freewheel. The tops of the teeth have grooves down the center, I'm sure there's a technical name for it, but I call these freewheels "split top teeth" freewheels. Likely the tooth design was to facilitate shifting. They were common on Schwinns and Peugeots in the 1960s and 1970s, and likely came on many other brands as well. These freewheels were a constant problem at the used bike shop I worked at.
Sometimes these freewheels work fine, sometimes they have a propensity to behave like yours. And then you get "ghost pedal" and things aren't fun at all. Replace the freewheel with one without split top teeth and you'll be fine.