This is correct. Specifically in the US, the conference of bishops has determined that Catholics are to abstain from red meat and poultry on all Fridays during Lent, but may substitute an act of penance on Fridays outside of Lent instead if they wish at their individual discretion. I think that is a fairly common arrangement in many countries. Truthfully, though, in practice, many Catholics are only aware that they are supposed to abstain from eating red meat and poultry during Fridays of Lent, and are not aware about the act of penance or any other requirements outside of Lent, so that contributes to some confusion about things in the general public.
So, why is fish not prohibited? Well, one has to understand that for Catholics, fasting and abstinence is not about being vegetarians for a day or about foods being clean or unclean (All foods are clean for Catholics as per the New Testament of the bible). Rather, these occasions are about self-denial and recalling and uniting each individual to the suffering of Christ, growing closer to God. It's not that vegetables are better than meat or whatever in some moral sense, it's just about people's relationship with God.
So, largely, the fact that this or that is chosen as a way of expressing that concept and working to that end is largely arbitrary -- if the Pope or the bishops wanted to, they could say all Catholics are going to abstain from candy and alcohol instead, or all Catholics are going to skip a meal, or whatever. It's more about the spirit of the action than the nature of the action itself.
I actually think there might be something to the urban legend that said that it used to be that fish were prohibited on Fridays along with the rest and it was changed because fishermen were struggling to make a living. I can't verify that for a fact, but I remember hearing it somewhere. If it did happen, it would be in accord with the theology and the intent of the thing. If abstaining from fish was putting people out of work, it would be charitable for the Church to exempt fish from the fast, because the fast really wasn't about the fish in the first place.