I've only fly fished for Carp. They're like the Bonefish of the West and are known to break an 8wt rod like a toothpick.
Does the game support fly fishing?
No fly fishing yet, still in early access, plan is to launch with regular carp fishing, then add on from there. This video is from the Training Lake, but they also just added in St. John's and next update (i think) they are adding in some place in the Canary Islands. They've been quite receptive to ideas and additions, I know fly fishing has been mentioned and iirc it's something down the line.
Fish atm are Common, Mirror and Leather Carp, also Bream. They have showed the models for tench as well.
I've put +130 hours into it so far (most during their 6 week tourney) and well things have come along very well and to their credit, if something doesn't seem quite right and people mention it, they make sure to get it a heck of a lot better.
I never caught Carp here for any other purpose than catfish bait, so not 100% certain on how accurate the carp fishing stuff is, although I don't hear any other folks saying about it being unrealistic in any regard. Whether you carp fish or not it is still a fun game.