I was really hoping there would be a category solely focused on fishing as it is my favorite thing to do in the game, and I thought catching all the legendary fish would be a great thing to speedrun! Do you think it's a possibility that could be an added category in the future?

There is a chance of being added but it lost out in recent votings for popularity. There isnt likely to be another vote for a while.

You can always keep track of your times though and try to better the times.