In some of the photos it seems that the animal has horns, though subsequent reports are that the “horns” are actually displaced bones protruding from the rotting carcass. I can’t think of any particular “Mormon angle” for this particular beast, but since we’re in the neighborhood… there are a few things to be said, briefly, about figurative language, Mormons, and sea creatures of uncertain taxonomy.
We might start with the many animal classifications derived partially from the Greek root “mormo,” some of which I have described before. Washed up “sea monsters” also figured prominently in the rise of the Mormon “octopus” or “devil fish,” which I wrote about earlier this summer (1, 2, 3, 4). In this connection—and while we’re on the subject of (pseudo-)cryptids by the seashore—Jenny Hanivers have also occasionally been called “devil fish,” but I am unaware of any Mormon connection.  In contrast, an 1890s theatrical poster announcing “The Mormon Senator and the Mermaid” has a clear Mormon connection, but I have been unable to locate further information.  Half a century earlier, PT Barnum alleged that an associate involved in the exhibition of his “Fejee Mermaid,” one Levi Lyman, “afterwards became a prominent Mormon, and removed to Nauvoo, where he died.”  I am disinclined to take Mr Barnum at face value, so I’ll file this as “awaiting confirmation.”
In the present case, however, I think the most obvious stepping off points are either the Bear Lake Monster, whose history is well summarized already, and this little guy, who appeared on the cover of The Daily Graphic’s 1883 Nov 01 issue :
I don’t know how intentional the “Chinese-ness” of the dragon is, but it could show another attempt to “Orientalize” Mormons at a time when anti-Chinese sentiment was high (the Chinese Exclusion Act had been passed the previous year). Note also the horns. A small subset of the corpus of “Mormons have horns” literature is comprised of non-humanoid, metaphoric representations of Mormons; this image is the earliest of that sort which I have identified with unambiguous horns.
Back to Almería: a common conjecture among commenters is that the specimen is an oarfish, which seems plausible to me. Oarfish belong to the order Lampriformes, which is phylogenetically and etymologically NOT related to lampreys , which does not stop me from pointing out that in 1911, Henry Alfred Lewis—a frequent contestant in this series—wrote of alleged Mormon financial prowess:
Once, as a boy, I saw a Lake Erie whitefish to which a lamprey eel had fastened. The eel was fat and sleek; the fish showed thin and worn. A fisherman told me that it was only a matter of months when the eel would have drained out its life and killed the fish. The country is the whitefish, the Mormon Church the eel, and the question for you to consider is, “How soon will the country die?” 
Finally, there is Leviathan, the sea monster of the Old Testament. In 1843 Joseph Smith wrote a letter to the (Mormon) Times and Seasons describing (unfavorably) various newspapers as lions. “Among these was a great lion”—later identified by BH Roberts as the New York Herald—with, among other attributes, claws “like the claws of a dragon” and ribs “like those of a Leviathan.” 
An 1854 piece addressing, among other issues, the teachings of “Romanist” and “Mormonite” workers, described the “Mormon church” as “a well-favored harlot” and a “leviathan.”  The idea showed up again in the 1880s. One DL Leonard (1886) alluded to Job 41:1-2 when he wrote of Mormon leaders that “the hook had not yet been put in the nose of the Mormon leviathan.”  The following year a Leonard Bacon lamented that federal anti-polygamy efforts were ineffective to the point that “Mormonism laughs at such expedients, like leviathan at the shaking of a spear.” 
In 1919 a G Seibel mocked the Book of Mormon by noting that “…the work adds nothing to human knowledge except the names of two animals, ‘cureloms and cumoms,’ hitherto unknown to Zoology. Along with Leviathan and Behemoth, these creatures would prove drawing-cards for any menagerie.” 
The Mormon leviathan has reappeared more recently in D Michael Quinn’s description of religion in the American West:
If any symbol captures the religious West, it’s the symbol of a giant aquarium—God’s aquarium. Throughout the wide spaces in God’s western aquarium, there are schools of familiar (but easily startled) denominational species, there are slow-moving crustaceans, there are religious exotics from the depths and an occasional shark, there’s the Mormon leviathan, and unchurched plankton are floating everywhere. 
 The “Mormon devil fish” seems to have referred unambiguously to a cephalopod, either an octopus or a squid.
 The poster is quoted in an article on slang. The full poster quote is “The Limit at Last! [¶] ‘The Mormon Senator and the Mermaid.’ [¶] Jags of Joy for Jaded Johnnies.” The author introduced it as “a theatrical poster, widely displayed in New York while I was there, bore this alluring inscription….” The author’s interest is in the slang meanings of “the limit” and “jags.” William Archer, “The American Language,” Pall Mall Magazine 19 (London, 1899): 195 (188-196).
 Phineas Taylor Barnum, The Life of P.T. Barnum: Written by Himself (New York: Redfield, 1855), 241.
 No illustrator listed, “What Shall They Do to Be Saved?” drawing, caption: “The Edmunds Law is a failure — What shall Uncle Sam do next?” The Daily Graphic 33.3294 (1883 Nov 01 Thu): 1 (cover), image courtesy of “Old Fulton New York Post Cards,” fultonhistory.com. Uncle Sam is at left holding a sword but entangled in tape with the words “Red Tape” and “Edmunds Law” repeating in sequence. An eagle is perched on his head. Sam confronts, at right, what looks like a five-clawed Chinese dragon with “Mormonism” written along its side. Behind the dragon, and imprisoned by it, are ten young, White women in attitudes of distress such as praying, crying, clutching hair, and so on. On the cliff wall behind the women is written “Degradation worse than slavery.” The background shows mountains, a valley, and some indistinct buildings.
 If Wikipedia is to be trusted, “Lampriformes” is from the Greek lampros = bright, and “lamprey” is from lampetra = stone-licker, from Latin lambere = to lick, and Greek petra = stone.
 Alfred Henry Lewis, “The Viper’s Trail of Gold,” Cosmopolitan 50.6 (1911 May): 827 (823-833). Lewis had used a related metaphor a few months prior to describe US Senator Elihu Root: “Aside from darting, wide-mouthed qualities of a pickerel voracity, Mr. Root is as slippery as any eel. All who sought to fix him to either a principle, a course, or a fact, found him like an eel—as elusive as any shadow. Like the eel, too, in whatever of politics or business engaged him, he lay hugging the bottom. They had to draw off all the water in the pond before locating lamprey Root—draw off all the water and then poke about in the ooze.” Alfred Henry Lewis, “Root,” Cosmopolitan 50.2 (1911 Jan): 256 (251-260).
 BH Roberts, editor, History of the Church… 5.275. Also in 1843 Smith wrote another letter which asserted that “the powers of righteousness and truth” associated with Mormonism were not “the lions of the land, nor the leviathans of the sea, moving among the elements, as distant chimeras to fatten the fancy of the infidel; but they are as the stone cut out of the mountain without hands, and will become a great mountain, and fill the whole earth.” Joseph Smith, letter to James Arlington Bennett, 1843 Nov 13, Nauvoo, IL, as quoted in No author listed [attributed to Charles Mackay], The Mormons: or Latter-day Saints, with Memoirs of the Life and Death of Joseph Smith, the “American Mahomet” (London, 1851), 116.
 “The characters which I have described are a part of the real Babylon, ‘the great whore,’ which ‘sitteth upon many waters.’ The Mormon church is ‘a well-favored harlot.’ Therefore, ‘Come out of her, my people, that ye be not partakers of her sins.’ This Babylon is become the habitation of devils, the hold of every foul spirit, and ‘a cage of every unclean and hateful bird.’ And if the Lord had not predetermined to keep his people from being led on to their destruction by this leviathan, they also would have filled up the measure of their iniquity. But the Lord has in mercy commanded his people to ‘come out of her.’” Author identified as “A Watchman,” “The People Against Whom the Lord Has Indignation for Ever,” The Gospel Standard 20:217 (London, 1854 Jan): 19 (18-21).
 “For the Edmunds Bill was yet almost a year in the unknown future, and so the hook had not yet been put in the nose of the Mormon leviathan. The theocracy was still proud and defiant. And well it might be; for after long conflict with the Federal Government it was unharmed, and had more than held its own.” DL Leonard, “Five Years in Utah,” The Home Missionary 59:6 (New York, 1886 Oct): 228 (225-230).
 “There was hope that acts of Congress against polygamy, and prosecutions before United States Judges for marrying more wives than one, would break up the harems of the hierarchy, and open the way for Christian civilization to displace the bastard Mohammedanism invented by Joseph Smith. But Mormonism laughs at such expedients, like leviathan at the shaking of a spear.” Leonard Bacon, “What Are You Going to Do About It,” in Jennie Anderson Froiseth, ed., The Women of Mormonism, or The Story of Polygamy as Told by the Victims Themselves (Detroit, MI: CGG Paine, 1887 [1st ed, 1881]), 306 (303-311).
 George Seibel, The Mormon Saints: The Story of Joseph Smith, His Golden Bible, and the Church He Founded (Pittsburgh: Lessing, 1919), 33.
 D. Michael Quinn, “Religion in the American West,” in William Cronon, George Miles, and Jay Gitlin, eds, Under an Open Sky: Rethinking America’s Western Past (New York: Norton, 1992), 164 (145-166).