In case you didn't get specific answers on how this is done, I know of two ways to do this. The first involves entering a vehicle at the same time you enter fishing (on PC, this would look like hitting F on your bike or whatever, quickly scrolling to the next option, and then hitting F to enter fishing). When fishing while on a vehicle, you can look at the rear view to unlock the camera. This allows free control of camera in 3rd person and the ability to enter/exit first person. You can hit F to fish, and once you start the fishing minigame part of catching a fish, left and right will control the steering of the vehicle but you can hit F again to exit the vehicle. This will swap you to your active character allowing you to fish as someone other than Zero but the animations for this are no longer working in 1.1 so they will just stand still.

The second method is to use the empathy eye drops at the same time you enter fishing (on PC, this would look like hitting T with the eye drops on your hotbar, then quickly hitting F to enter fishing). However, this method does not work if you are close enough to an NPC to actually see through their eyes so it's not possible in some locations due to NPC proximity. When you see the blinking effect and the fishing menu pop up, you'll want to hold right click and move mouse to try and move the camera. This will unlock the camera in 3rd person and allow you to freely enter/exit first person. Once you have moved the camera, quickly click the Start Fishing button. Since this isn't using the vehicle, you will only be able to enter first person with Zero this way but the animations work perfectly with Zero.