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Fruitcake is an insulting word for someone who you think is strange or crazy (the Macmillan Dictionary).

Why does the word have this meaning? What is the similarity between a strange person and a heavy cake containing dried fruit?

The answer may be connected to the expression "nutty as a fruitcake."

Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) has this entry for that expression:

nutty as a fruitcake Crazy, idiotic, as in Mary's nutty as a fruitcake if she thinks she can get away with that. The adjective nutty meaning "insane" was firt recorded in 1821; the similarity to fruitcake, which literally contains nuts as well as fruit, was first recorded in 1935.

Well, no—"nutty as a fruitcake" was recorded at least 23 years earlier, in "On Main Street: A Decline in a Noble Industry"—a rather silly take-off on "Who Killed Cock Robin?"—in the Saturday Evening Post (July 20, 1912):

"Assuming, doctor, please, that on the night in question—namely, as follows, the night of the ostensible death of the alleged Cock Robin—the defendant at the bar, Clarence Algernon Montmorency Sparrow, alias English Chip, had been eating a great number of things or else drinking them; and assuming that he did not wer the calm, reflectful cast of countenance of a man who is trying to decide whether h will have his eggs fried on one side or turned over—but on the contrary, that his hair was mussed and his eyes tolled in their sockets, precisely like a pair of white china doorknobs, only radically different; and assuming he had a breath like a brandy sauce, and that, as the English say, a man's souse is his castle; and assuming further that you are going to get a large and nourishing fee if you make the right answer; and assuming that the defendant's Aunt Jane, in Wilkes-Barre, was as nutty as a fruitcake, and his grandfather, in Oskaloosa, had to wear cotton in his ears to keep his brains from blowing his side-whiskers off—and that they were acutely maniacal sidewhiskers, as all sidewhiskers necessarily are; and assuming that an Irishman wears red suspenders because the old hen crossed the road to get on the other side to find out how old Ann is and lay an egg and a half in a day and a half; and assuming quite a large number of other things of the same general nature or entirely otherwise, as the case may be; and assuming—"

That inauspicious occurrence puts "nutty as a fruitcake" very close to the earliest days of "nuts" as slang term for "crazy." I haven't found any examples of "So-and-so is a fruitcake" (meaning a crazy person) of comparable age. It would not surprise me if "fruitcake" in the sense of "crazy" originated in the longer form "nutty as a fruitcake," where the tell-tale insignia of insanity were the nuts.

At least since 1950, fruitcake has appeared as a pejorative term for homosexual. J.E. Lighter, Random House Historical Dictionary of American Slang (1993) has this entry for fruitcake:

fruitcake n. {sugg. by nutty as a fruitcake; see NUTTY} 1. a crazy, eccentric, or silly person; in phr. go fruitcake to go crazy. [cited occurrences, from 1942 forward, omitted] 2. an effeminate male homosexual; FRUIT, [definition] 4—used derisively. [Earliest cited occurrence:] 1960 Hoagland Circle Home 232: I'm even looking for a fruitcakes' club, to use the sailors that are going to pour in here.

The complication here is that Lighter's fruit definition 4 ("an effeminate male; a male homosexual.—used derisively.occ. used collect.") dates to 1900:

1900 D[ialect N[otes] II 37: Fruit, n. ... An immoral man. {Reported from Tulane Univ.}

The cited glossary from Dialect Notes is Eugene Babbitt, "College Words and Phrases" (June 1900), and it is noteworthy for two things that Lighter doesn't mention. First, the "An immoral man" meaning of fruit is the tenth of ten definitions listed:

fruit, n. 1. A person easily influenced. 2. One easy to defeat. 3. An instructor whose course is not exacting. 4. An immoral woman. 5. A good fellow; a trump. 6. A social function. 7. A girl whose acquaintance is easy to make, 8. A tag which has been removed by a sophomore from a freshman's shirt. 9. A disagreeable person. 10. An immoral man.

Second, the relevant definition is attested at only two of the 45 universities that reported slang meanings of fruit: Tulane University in New Orleans, Louisiana; and Wyoming Seminary, in Kingston, Pennsylvania. These states are not particularly close geographically.

In any event, the 1900 instance of fruit as a pejorative term for an "immoral" (that is, implicitly, gay) man is somewhat older than the earliest (1912) instance I've found of "nutty as a fruitcake" as an idiomatic expression for a crazy person. Since fruitcake itself doesn't emerge as an insult term for a homosexual until much later, it seems clear that the original emphasis in "nutty as a fruitcake" is on the nuts, not the fruit. But the two have had a somewhat complicated relationship in the vocabulary of a hostile outside world.

Fruitcake seems to be a shortening of earlier phrases such as "nutty as a fruitcake". The oldest form I can recall is "nuttier than a fruitcake"

Wikipedia has more on this. Pinning slang down to a certain origin is next to impossible. Fruitcake is not a description I would use for anybody as its meaning might not be clear.