Here are some additional human given names that are in current general use for certain animals. I don't think that these have been explicitly identified by other answerers/commenters.
Widespread General Use
billy goat [from William]
jackdaw, jackrabbit, jack mackerel, jack salmon, jacksmelt (common name, both sexes) [from John]
jenny wren [from Janet]
john mule, John Dory (common name of the fish, for both sexes) [John]
magpie [from Margaret plus pie in the sense of "pied {that is, blotched black-and-white} crow"]
molly mule [from Mary]
nanny goat [from Anne]
poll parrot [from Mary, according to OED via Word Origins, but not gender-specific in use]
tomtit (common name, both sexes) [from Thomas]
Regional (and Possibly Obsolete) U.S. Use
These names appear as entries in Harold Wentworth, American Dialect Dictionary (1944) but are probably not in widespread use today:
Bessie cow [from Elizabeth] ("A cow." The entry cites a 1942 article in the Saturday Evening Post.)
biddy (for a hen or young chicken) [from Bridget, in its perhaps unrelated meaning of "hired girl" or "elderly woman"; "biddy" the chicken dates to 1601, and Merriam-Webster's says that the name is "perh[aps] imit[ative]," while "biddy" the working girl dates to circa 1861 and is "dim[inutive] of the name Bridget"]
bubbly-jock [from John] (The entry says, "A turkey gobbler. 1930 central Pennsylvania mountains. Still or recently used. 1934 Chiefly Scottish. Webster's.")
jack [from John] ("A male animal. 1923 northwestern Missouri Not used in mixed company. Male, n., is used. 1934 = male of certain animals. Webster's.")
jessie [from Jesse] ("A 'critter.' 1942 Florida.")
Johnny, Johnnie [from John] ("A male animal. 1934 Local U.S. Webster's.")
Old Ned [from Edward] ("1904–1922 western North Carolina–eastern Tennessee 1936 southwest Missouri–northwest Arkansas 1941 1. Fat pork, bacon. 2. A boar.")
The strangest animal name entry in the American Dialect Dictionary is surely "Old Ned," which in some localities refers to certain delicious remains of the animal in question, though it may also refer to a whole (and presumably live) boar.
False Alarms
Two bird names that don't qualify because they are imitative of the bird's call, rather than being based on human given names:
bobwhite
chuck-will's-widow
Two designations that don't make the cut for other reasons:
pollywog [probably comes not from the name Polly but from poll (in the sense of head) and wiglen (to wiggle)]
teddy bear [doesn't apply generally to real bears]
And two involve the reverse case of names that arose from animal-specific words in other languages and later became common popular given names in English owing to the appeal of the animal:
mavis [derived from Anglo-French mauviz, meaning "song thrush"]
robin [derived from a root akin to the Danish dialect word robijntje and the Frisian word robyntsje, meaning "linnet"]
A Note on Polite Avoidance of Certain Animal Names
I was struck by the remark in the American Dialect Dictionary that, in the 1920s in northwestern Missouri, jack was not used to refer to a male animal "in mixed company." It turns out that, across much of the U.S. South and as far north as Nebraska, Kentucky and West Virginia, some rural folk considered various male animal identifications to be vulgar. None of the forbidden nouns would cause much of a stir today: "Bull, boar, stallion & jack are not used in mixed company, although buck (male sheep or goat) & crower (rooster) are." But squeamishness on this point led to such euphemisms as male ("Any male animal kept for breeding purposes"), stock-male (bull), male-brute (bull), male-cow (bull), male-hog (boar), and male-pig (boar), which were in use for at least the first four decades of the twentieth century in some parts of the United States. The only animal designations (by gender) I can think of that prompt similar recourse to polite circumlocutions today are cock and bitch.
Best Answer
To take one of your examples, doe is from old English deon, to suck, and was used to describe a female animal of a number of kinds. buck is from old English bucca, a he-goat. The unisex rabbit is from Flemish. So the words buck and doe appear originally to have meant the male and female of any animal, and have presumably become specialised in referring to deer, rabbits, and certain other kinds of animals.
Ram and sheep are both derived from old English, in the case of ram a word meaning fighting sheep. (Ewe is from Latin.) I'd hazard a guess that in old England there were fighting sheep (males) and sheep (all the others, including wethers).
Hope this helps (all this from the Collins English Dictionary, except the suppositions and guesswork).