I simply had a pig on a lead and used my mouse wheel to select a fishing rod. I right clicked as normal to use the fishing rod and the rod worked as normal. When I right clicked to bring the fishing wire in, another fishing wire was created, again and again (see pic).

Step by step
1. Place a fishing rod and a lead on your hotbar
2. Right click on an animal (in my case a pig) to attach it to the lead
3. Use your mousewheel to scroll to your fishing rod
4. Right click with the fishing rod mutliple times
5. You'll see multiple fishing wires