Maybe the dog or horse with coat of the color that we call flea-bitten has been at some time infested with fleas, but, whether so or not, that is unessential.
The color term originated back in the latter half of the sixteenth century.
Some discerning groom or dog fancier saw that the reddish flecks on the coat of his lighter-colored animal were very similar in appearance to the reddish marks left on his own hairy arms by the bites of fleas.
So what more natural than to describe the coat of his animal as flea-bitten?
It saved time searching for a more definite color; easier to think of than, say, mottled gray or speckled sorrel.