I mean, it's not like there is a tuna vegetable or animal that it can be confused with.
Learn English – Why do Americans say “tuna fish”
american-english
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I did some searches in the Corpus of Contemporary American English and compared the results to similar searches in the British National corpus.
What I found was that overall, in American English there was a 7.9-to-1 ratio of don’t to do not. With breakdowns by type:
SPOKEN 19.6
FICTION 17.9
MAGAZINE 7.5
NEWSPAPER 7.7
ACADEMIC 0.5
TOTAL 7.9
In British English overall, the ratio was 4.4-to-1 in favor of don’t, with breakdowns by type:
SPOKEN 56.9
FICTION 16.8
MAGAZINE 4.1
NEWSPAPER 3.4
NON-ACAD 0.9
ACADEMIC 0.2
MISC 0.9
So, if it is reasonable to conclude anything from this data, it is that Americans overall use don’t about twice as frequently as the British, but the British use don’t in speech about 2.9 times more frequently than Americans. In any case, these are not big enough ratios to be noticeable by anyone not counting every incidence.
Þære spræce and þære ácsunge
You suggested perhaps the 1800s for the origin of this use: I’ll see that bet and raise you a millennium. Not an Americanism but perhaps a Wessexism, for some one thousand one hundred and thirty years ago in ᴀᴅ 885, King Ælfred the Great wrote not only:
Hit is þeaw þære spræce and þære ácsunge.
but also:
Mid ascunga.
Much more recently, Shakespeare wrote in King Henry VIII:
Bestowing on him, at his asking, The Archbishopricke of Toledo.
And again in Coriolanus:
Yet dare I neuer Deny your asking.
When Tennyson wrote Tithonus, he deliberately used this archaizing word to lend an air of antiquity to his verse:
Then didst thou grant mine asking with a smile.
This deliberately hearkens back to olden times, where such things as this from the 1410 Pynson translation of Bonaventura’s Myrrour were readily found:
Our lady··answereth sadly and shortly to theyr askynges.
Almost all such uses of asking save for Ælfred’s are now considered archaic or obsolete by the OED. The King’s sense of the word was “the action of putting a question, interrogation, inquiry”, and this is the one still employed in yours for the asking.
Inquiries quickly become requests, and Shakespeare’s Henry VIII use (“the action of requesting a favour, gift, etc.; praying, begging”) is also still considered current. However, his Coriolanus use (“a petition, prayer, a supplication”) is now archaic.
There is also the regional use of an asking meaning the publication of marriage “banns” (proclamations of marriage), but this is very limited in distribution.
Summary
Even though askings as a count noun has existed since time immemorial, it is now sufficiently obsolete that most spellchecking programs will flag it as an error in Modern English.
To most people, “What’s your ask?” and “Did he send in his asking yet?” sound like nothing but annoying business jargon instead of like forgotten pieces of English dug up from some ancient graveyard.
All citations are from the OED.
Best Answer
I agree with you that it does seem redundant. However, this is common with other kinds of fish as well. Many people say "codfish" instead of "cod". Here is a recipe for "trout fish" croquets.
This convention has important meaning to a huge number of fish names: catfish, lionfish, swordfish, sunfish, cowfish, etc.
Also, it provides extra clarification for someone who wouldn't know what a "tuna" or a "cod" is otherwise. Anyone learning English as a second language will probably learn the meaning of "fish" early on, but may not know the more specific names.