Wordorigins.org

The metaphor at work in the phrase spitting image is that of a person spitting out an exact replica of themself. This metaphor dates from the first half of the 19th century. From Knapp & Baldwin’s 1825 The Newgate Calendar:

A daughter,...the very spit of the old captain.

The phrase spitten picture dates to 1878. From William Dickinson’s A Glossary of Words and Phrases Pertaining to the Dialect of Cumberland:

Spitten picter..., a strong likeness.

Finally, spitting image is in use by 1901 when it appears in Alice Rice’s Mrs. Wiggs of the Cabbage Patch:

He’s jes’ like his pa—the very spittin’ image of him!

There are several false explanations circulating about this phrase, probably because the metaphor is not an obvious one. One is that it is an alteration of spirit and image, from the belief that people who looked alike shared a soul. Another is that it is a variant on splitting image, referring to a person being split into two identical persons. Both of these are incorrect, although use of splitting image is almost as old, dating to 1880, indicating that the spitting metaphor was not understood by everyone in the 19th century. These are after-the-fact explanations, unsupported by evidence. Instead, the spit is just what it sounds like, saliva.

(Source: Oxford English Dictionary, 2nd Edition)