why is it "a fisherman" wouldn't the correct translation be "the fisherman"?
Indeed it is the definite article in the Italian and should therefore be "the".
Why is it "fa" here and not "è"?
They are interchangeable when describing professions.
You can use 2 constructions: Mio papà fa il dottore OR Mio papà è un dottore (both are translated as: My dad is a doctor)
Can you use this constructions with things that aren't strictly professions? Say for example, 'faccio la studentessa'.
So if you had used "è" here it would be translated as "My father is the fisherman" but because it's "fa" it's "My father is a fisherman"?
that's what it seems like
so in this case il pescatore means a fisherman (not a specific but as a role)
Ho scritto: "My fther is fisherman": Anche in questa risposta c'è confusione! L'articolo indeterminativo "il", viene sostituito, arbitrariamente, con l'articolo indeterminativo "un".