Certainly there is a nuance of irony over the extreme contrast between "fish cooking" and "a doctor (including a physician, a surgeon, etc.)".
I think that the basic part of the contrast is "to cut flesh" or "to execute a surgical operation".
The speaker thinks that there is "a surgical operation" as one of the main part of the doctor's work. As an amateur's understanding, the speaker seems to think that "surgery", "cutting flesh" or "dissecting" are the same thing as a whole.
On the other hand, I think that the speaker assumes that cleaning fish (removing the scales and guts from fish) and filleting fish (removing bones from fish), that are similar to a surgical operation, have to be done prior to cooking them.
As a whole, the speaker thinks and says ironically and ignorantly that the student cannot become a doctor because he cannot even cook fish.