It’s nothing to do with the Feeding of the Five Thousand — it’s a very old Roman Catholic discipline, a day of abstinence from flesh meat. It used to be ecclesiastically compulsory in the Roman communion, and throughout Lent, into the first half of the 20th century with more or less rigour. I believe that the Eastern Orthodox not only still keep it, but also go without milk, butter and cheese.
In answer, yes and no. I don’t eat meat on Fridays, unless I will offend a more Protestant friend or relation, but not necessarily fish.