Are sharkfin melon shoots really edible?
According to a number of sources, the shoots are indeed edible, however all references I found were to "young" shoots, so I don't know about the mature vine itself. 

The Curious Gardener says


  It's not just the melon that is edible on this plant; the flowers,
  leaves, shoots and seeds are also eaten.


Although the flowers, leaves and shoots are mostly used raw as salad greens, 
The Food and Agriculture Organization of The United Nations adds:


  The young stems (called runners), and also the flowers, are eaten
  as a cooked vegetable.


This
North Carolina State University Article lists a number of cucurbits with edible shoots, including Chayote, Bitter Melon and a few types of Gourds.