Oxford English Dictionary:

3. the very spit of, the exact image, likeness, or counterpart of (a person, etc.). Also, the (dead) spit of. colloq.

1825 KNAPP & BALDWIN Newgate Cal. III. 497/2 A daughter,..the very spit of the old captain. 1836 T. HOOK G. Gurney I. 202 You are a queer fellow — the very spit of your father. 1885 HALL CAINE Shadow of Crime II. xxvi. 129 A brother..the spit of hissel’. 1886 MACQUOID Sir J. Appleby III. x. 143 This young chap has got his dear grandmother’s eyes, why, he’s the very spit of her. 1901 E. W. HORNUNG Black Mask 37 I’ll chance you having another ring..the dead spit of mine. 1921 ‘K. MANSFIELD‘ Let. Sept. (1977) 232 One of his [sc. Cézanne's] men gave me quite a shock. He’s the spit of a man I’ve just written about, one Jonathan Trout. 1936 M. DE LA ROCHE Whiteoak Harvest v. 98 Easy for a boy to look like his grandmother. There was Renny — the spit of old Gran! 1953 A. UPFIELD Murder must Wait xvii. 154 The son’s the dead spit of the old man. 1966 [see GRAMP].

TakeOurWord.com:

Spitting image, yet another term for “lookalike”, was originally spit and image. Spit meant “likeness”, so the term uses redundancy to bring home its meaning. It is first recorded in the late 19th century as spit and image, and by 1925 it had taken the current form. The Oxford Dictionary of English Proverbs, however, notes the saying “as like one as if he had been spit out of his mouth” as having been recorded in 1400, so the notion behind spit and image was already alive in the late Middle Ages. An apparently spurious etymology of spitting image has the term coming from spirit and image, and another from splitting image.

The Columbia Guide to Standard American English:

The key to this puzzling cliché is the word spit, which seems to have been an early-nineteenth-century dialectal noun meaning “an exact likeness or counterpart of a person or thing,” as in He’s the very spit of his father. The question is, how did spit develop that sense, dialectal or not? The Oxford English Dictionary concludes that it developed somehow from the noun spit, meaning “expectorated saliva,” but no really satisfactory explanation of the semantic change involved has thus far turned up. In the late nineteenth century the full cliché developed: He’s the very spit and image of his father, followed by the folk etymology that replaced spit and with spitting (or spittin’ or spitten) image. The phrase has a rather countrified air about it, and it is not likely to be found today in either Formal writing or at the higher levels of speech.

My only elaboration is to say that “the term uses redundancy to bring home its meaning” does not tell the whole story. After the Norman conquest of England in 1066, legal writing developed the custom of indicating the significant nouns first in their shorter, Anglo-Saxon form, then in their longer French/Latinate form, e.g., “last will and testament” where “will” and “testament” mean the same thing, but one was the term understood by the locals while the other was in the language of the conquerers. (I can’t find the passage where I first read this, but I believe it was in Bruce Benson’s Enterprise of Law.)

My guess is that this habit of short+long or local+Norman seeped out into the non-legal phrases of the language.