That's pretty fun minigame.
At first I lost, but second time could beat it. Once even caught a yellow fish.
Cool music and sound effects.
Viewing post in Soporo: Fishing In Ruin Demo jam comments
The require you to wear down their stamina like any other fish before catching. They have large stamina pools, but they actually don't regenerate stamina any faster than the blue fish, meaning that after you've fought him for a while, he's weakened and you can reel him in. He puts up a hell of a fight before that though.