I’m looking for the earliest story with clones (genetic duplicates of people created via technology and grown from cells/embryos) in it. Just how old is the trope?
Brave New World 1931 (published 1932, Aldous Huxley).
The Bokanovsky process which was enacted before birth in which the newly fertilised egg was forced to divide an arbitrary number of times to create up to 96 identical copies.
Gamma, Delta, and Epsilon ova are removed from their incubators so that they may undergo Bokanovsky’s Process. “One egg, one embryo, one adult—normality. But a bokanovskified egg will bud, will proliferate, will divide. From eight to ninety-six buds, and every bud will grow into a fullsized adult.
Although never referred to as "clones" the copies would be identical in appearance and conditioning and be assigned to their predestined roles dependant only on their genetic stock.
Definition of clone taken from American Heritage Dictionary.
The Tissue-Culture King 1926 Not full clones but cell line tissue cloning. Basically a tribe that worships their chief and elders switch to preserving cell lines. Which greatly expands the freedom for the original king. Full story https://archive.org/details/Amazing_Stories_v02n05_1927-08_017/Amazing%20Stories%20v02n05%201927-08%20017?view=theater#page/n35/mode/2up
Description from Wikipedia https://en.m.wikipedia.org/wiki/The_Tissue-Culture_King
The Tissue-Culture King (1926 in Cornhill Magazine and in The Yale Review, reprinted 1927 in Amazing Stories and many times afterwards) is a science fiction short story by biologist Julian Huxley.
The story tells of a biologist captured by an African tribe. It incorporates the idea of immortality based on reproduction from a tissue culture and genetic engineering...
Another near miss is Le Singe, 1925, Maurice Renard and Albert Jean. Translated as Blind Circle. Synopsis from here:
Renard’s most detective-like sf narrative—perhaps patterned on Poe’s The Murders in the Rue Morgue—was published in 1925 and carried the title Le Singe (curiously translated as Blind Circle, 1928). It was co-authored with Albert Jean and, like most of Renard’s fiction, originally appeared in feuilleton format (in 62 episodes) in a Parisian pulp journal. Its plot concerns a police investigation of the mysterious presence of multiple dead bodies—all of the same person, a certain Richard Cirugue. After innummerable twists and turns in the narrative, the mystery is finally solved: Richard is actually still alive. Using his recent discovery of an electrolysis procedure to duplicate (but not animate) animal and human tissue, he had fabricated lifeless clones of himself and scattered them around Paris as a publicity stunt to attract investment capital for his scientific research. In an even more bizarre twist, Richard subsequently dies (for real) but his spirit manages to survive and ultimately inhabits the cloned body of his brother, whose wife he lusts after. Needless to say, apart from its somewhat Frankenstein-like motif of creating synthetic bodies out of a chemical vat, Le Singe is less a science-fiction novel than, as one critic has said, a "comedy of manners, sex, [and] mystery in the manner of Gaston Leroux" (Bleiler, 620).
(emphasis is mine.)
Note that discussion of clones and cloning, by that specific term, is common and frequent in nonfiction journals of agriculture and plant science of the 1890s-1930s.