Q:

Fish Irregular Plural Forms

Several species of fish have names that are both singular and plural form.  These include cod, flounder, salmon, and trout, they are used to describe one fish or ten.  Does this stem from fish being both singular and plural?  Was the irregular plural form passed along to the species of fish?  

A:

I expect an answer to your question will be difficult to come by.   Many fish names form regular plurals (Bluegills, guppies, sardines), and many of the irregular plurals are fairly modern usages, so someone will have to account for the regular plural disappearing. 
As You Like It, Act II, Scene IV.  Touchstone says:

... and I remember the wooing of a peascod instead of her, from whom I took two cods and, giving her them again...

From The Sportsman's Dictionary: Or The Gentleman's Companion: for Town and Country. ( 1800) in a review of various rivers, one in particular:

well stored with gudgeons, dace, flounders, perch, pike, and some carp and trouts. 

The American Fisheries Society has helpfully compiled a list of the proper plurals of fish names in A Guide to AFS Publication Style.   Go here and look for Appendix C.

A:

As I mentioned late last month in answer to the identical question, I have long suspected that with a few exceptions, species of fish (and sometimes fowl) destined for human consumption are often treated as mass nouns in Modern English, mostly because they are seen as commodities like flour or sugar. 

bass, halibut, herring, carp, perch, trout, cod, salmon, tuna, pike, mackerel, flounder

One sardine won't make a meal even for a kitten, so it's hardly a surprise sardine has a plural, but krill are so small they don't even rate a singular. 
Tilapia is a species relatively recent to the seafood counter, likely the reason the style guide of the American Fisheries Society (scroll down to the appendix) lists it among those that can form an s-plural or not. 
Most crustaceans form s-plurals even though delicious — lobsters, scallops, clams, mussels — but while crab and shrimp may be plural crustaceans, i.e., doing their natural thing in the ocean, they are almost always mass noun shellfish, i.e., playing a crucial role in gumbo. 
Any fish whose name is a compound ending in -fish stays -fish in the plural whether you eat it or not. 
The men in my family of my father's generation enjoyed dove and quail hunting, so these two birds were always mass nouns at home, but I noticed that classmates who never ate them would use s-plurals.  My brother's friends, at least those who got BB guns for Christmas, hunted squirrel, though since the Depression I don't think they landed on anyone's table.  When one heard from old-timers that back then armadillos were called "Hoover hogs," then the question arose, "Can you eat armadillo?"  Ask or answer the question often enough in the affirmative, and I can easily see how this mass noun usage could in some cases transfer to any mention of the creature in question. 

A:

Fish do not seem in any way a special case, but the zero plural is common for animals.  And we tend to distinguish more different kinds of fish than sheep, or deer (see what I did there?) 
The zero plural as such is an interesting phenomenon.  The American Heritage Dictionary has the follwing interesting note:

In English, plurals of nouns are normally indicated by the ending -s or -es, or in a few cases by -en, as in children and oxen.  Some vernacular varieties of English do not use plural endings in measurement phrases such as three mile and ten pound. 
This zero plural has a long history and was not formerly as socially stigmatized as it is today.  It appears in literary works dating from the Middle English period to the present day, including works of dialect writers, such as this example from Mark Twain's Huck Finn: "The nearest white settlement warnt nearer nor four mile." 
In adjectival constructions even Standard English has no -s plural: a five-pound box of candy is acceptable, whereas a five-pounds box is not.  These adjective phrases derive from an -a suffix in Old English that marked plural adjectives.  This ending has long since fallen away, leaving behind the unmarked root forms. 
The absence of -s in the plural form of animal names (hunting for bear, a herd of buffalo) probably arose by analogy with animals like deer and sheep whose plurals have been unmarked since the earliest beginnings of the English language. 

The last paragraph specifically addresses animal names, and gives us the, in itself maybe unsatisfactory, explanation that these zero plural forms have existed since the dawn of English. 
Simply said, some words don't have a plural form, because they never had one. 
The Hartford Courant has an article about them, addressing three theories:

Game Theory: Many zero-plural words refer to animals that are commonly hunted. 

Rebuttal: True, we often do use the singular form when we talk about stalking animals ("We're hunting rabbit"), but not for all of our quarry, e.g., "We're hunting squirrels," not "squirrel." 

Grandfather Clause: The zero-plural words are very old words. 

Rebuttal: "Deer" and "sheep" did appear before the 12th century, but so did many animal nouns that form their plurals by adding "s," e.g., horse, dog, cat. 

Deutsche Doctrine: Many of the zero-plural words come from German, which rarely forms plurals by adding "s."

Rebuttal: True enough, "deer" and "sheep" do derive from German ("scaf" and "tior," respectively).  But "moose" comes from Algonquian (a Native American language) and "salmon" comes from Latin via the French. 

That last point is interesting, though.  The The American Heritage Dictionary mentions the possibility that plurals of a number of relatively new animal names in English where formed by analogy with existing (old) animal names.  Maybe the colonists who saw a moose figured it was a big deer, so two of them were moose, like two big deer.  Whatever their reasoning, it stuck. 
The conclusion from the Hartford Courant sums it up nicely:

English is mysteriously and delightfully fickle, an enchanted home where the buffalo roam and the deer and the antelope play — but the gophers and prairie dogs frolic, too.