Hate to be a stick in the mud, but I have to dispute this as insufficient sourcing. At this point, I’m more interested in this because of the alleged distinction being included at the Tuna article, too (where I’ll also raise this issue) than as an example at Pleonasm in a section that is about 90% original research anyway. I’m not sure it’s all that useful an example of anything because, while not ultra-rare, it is a sporadically occurring colloquialism, and more importantly because (as I show below) there’s hints that people who sometimes use “tuna fish” do not do so consistently, and even more evidence that some of those who do use it, use it in reference to the live animal (“also”, or “only” isn’t clear). If it’s taken this long to find even one dictionary-type source at all, that’s obvious evidence far from universal. It may not even be common any longer except perhaps among subsets of people. (But is it an age bracket? People from land-locked areas? Posts above indicate it has nothing to do with US vs. UK English, and everyone at least recognizes the phrase, or the Unix joke above wouldn’t work). I’m 43, I’ve lived in three English-speaking countries and five US states on both coasts and in the middle away from water, and I’ve never seen or heard anyone clearly make this distinction. The best I can confirm anecdotally is that I have heard people say “tunafish” (it always sounded like one word when I heard it), and as far as I can recall it was always in reference to food, including cat food, but it was always (AFAICR) in the [American] South and Southwest, far from anywhere there would be live tuna to refer to, so that could easily be false correlation.
“Tuna” without “fish” is used in both food and live-animal cases in every tuna-related context I can find, from canned fish (smoking gun evidence opportunity - find even one brand that says “tuna fish”!), to mixed products containing tuna, to fishing charter sites, to sea life sites, and so on. I cannot find a single tuna-bearing product that says it has “tuna fish” in it, not even cat food.
Interestingly, I did find “All About Tuna Fish”, a self-published pet food FAQ, but the name of the document is actually “all_about_tuna.PDF”, without “fish”, so even some people who use “tuna fish”, and use it in reference to food do so inconsistently. While it’s just one document, it’s worth looking at. It veers all over the place between “tuna” and “tuna fish”, including “canned tuna” without the “fish”.
More importantly, here is “tuna fish” in reference to the live animal (it turned up when I Googled “all about tuna fish” to find the above PDF again). I decided to just Google “tuna fish” without quotes: Some of the front-page results are for the live fish, including all four of the top image results, and the top YouTube result. Googling that phrase with quotation marks around it to limit the results more sharply still produced the same outcome.
The upshot being I think there’s not enough evidence that “tuna fish” is used consistently in any particular dialect, register, age bracket, context, or any other definable subset at all (and we have no source that it does), much less only to mean food, even less only to mean canned food, even if this might once have been true enough to be reported in one (and only one?) dictionary as a predicable distinction. Google turns up lots of evidence to the contrary. I.e., the one dictionary source is demonstrably not reliable in this case (probably because of limited data collection combined with age). The only conclusion that I can come to is that “tuna fish” is today like “puppy dog”, “kitty cat”, “pick-up truck” and “taxi cab” – it’s just something pleonastic that some people say and others don’t. “Puppy dog” and “kitty cat”, originating from pleonasm applied to “puppy” and “kitten”, are similarly inconsistent, in that people who use these phrases do not always use them, and do not reliably use them to only refer to something specific (here, juvenile pets - the terms are often applied to adult animals, just as we see “tuna fish” being applied to the fish as animals instead of food). With regard to your having added “tuna fish” and the same source to the Tuna article, where I’ve also disputed it (and removed the dialectal claim, since the source doesn’t support it), I have to note that neither the Dog nor Cat articles mention “puppy dog” and “kitty cat”.
Using “tuna fish” as an example of standard English idiom like “safe haven” pushes the issue even farther, and it isn’t supported by the dictionary source, even if that source were reliable on this and we didn’t have evidence of wildly inconsistent usage. The phrase survives at all only because older family members, who picked it up from even older ones (back to ca. 1881, when the word entered English from Spanish, according to the dictionary) from inland areas not used to various kinds of fish being readily available in the olden days, have passed it on to current generations, who move around in motor vehicles and thus make it hard to pinpoint exactly where it started. It’s just like “tin foil” for “aluminum foil” (I say “tin foil” all the time myself for this got-it-from-the-parents reason, even though it was supplanted by aluminum foil before I was born). It’s an unconscious language habit that will die off. While I have no doubt that some individuals, even entire families of them, actually do limit the phrase to canned tuna, well, so what? That’s not encyclopedic, even if an outdated dictionary also records the bare fact that the usage exists, but provides no context for that observation.