Here is a related line from Shakespeare that has become idiomatic:
The lady doth protest too much, methinks.
It comes from Hamlet and you can read all about it here. As the Wikipedia article suggests, it has come to mean that one can "insist so passionately about something not being true that people suspect the opposite of what one is saying."
I have heard it in many forms, such as "methinks the lady doth protest too much," "the lady protesteth too much," or more simply "she/he protests too much."
The horseback idiom appears older than the wishes one by around 100 years. The older one seems to be based upon a Latin phrase from around 400 AD, with this particular translation by Robert Burton in The Anatomy of Melancholy of 1621.
If wishes were horses, beggars would ride
It's listed in a proverbs book from 1721:
Complete Collection of Scottish Proverbs - Page 178 - James Kelly - 1721
19. If Wishes were Horses, Beggars would ride.
Interestingly, the other horseback idiom also appears in this book, along with a Latin phrase:
110. Set a Beggar on Horse-back and he'll ride to the Dee'l.
Lat. Asperius nihil est humili cum surget in altum.
More on that later.
Set a beggar on horseback, and he'll gallop
The John Trapp of 1650 appears to be a valid truncation, as three years later we find the full proverb:
"The Fourth Book of Dr. Francis Rabelais" - The works of Mr. Francis Rabelais, doctor in physick - Volume 2 - Page 238 - François Rabelais, Navarre society, London - 1653
Fryar Jhon began to paw, neigh and whinny at the Snout's end, as one ready to leap, or at least to play the Ass, and to get up and ride tantivy to the Devil like a Beggar on Horseback.
(Tantivy means at full gallop.)
And earlier, in a list of proverbs:
"Certaine Prouerbes ... of the English Nation in former Times, and some of this present Age." - Remaines concerning Britaine: But especially England, and the Inhabitants Thereof - Page 272 - William Camden - 1629
Set a beggar a horseback,and he wil gallop.
And Wikiquote attributes it earlier, from 1621:
Set a beggar on horseback, and he will ride a gallop.
Robert Burton, The Anatomy of Melancholy (1621), Part II, Section III. Memb. 2.
Checking the source, we find a Latin phrase:
And though by their education such men may be better qualified, and more refined; yet there be many symptoms by which they may likely be descried, an affected fantastical carriage, a tailor-like spruceness, a peculiar garb in all their proceedings; choicer than ordinary in his diet, and as Hierome well describes such a one to his Nepotian; An upstart born in a base cottage, that scarce at first had coarse bread to fill his hungry guts, must now feed on kickshaws and made dishes, will have all variety of flesh and fish, the best oysters, &c. A beggar's brat will be commonly more scornful, imperious, insulting, insolent, than another man of his rank: Nothing so intolerable as a fortunate fool, as Tully found out long since out of his experience; Asperius nihil est humili cum surgit in altum, set a beggar on horseback, and he will ride a gallop, a gallop, &c.
Asperius nihil est humili cum surgit in altum
Here's a likewise English-Latin translation from 1672, and English-Latin-French-Italian from 1678.
Wiktiquote attributes this Latin to the Alexandrian poet Claudianus (c. 370 – 404):
Here's the full passage from In Eutropium:
Asperius nihil est humili cum surgit in altum:
cuncta ferit dum cuncta timet, desaevit in omnes
ut se posse putent, nec belua taetrior ulla
quam servi rabies in libera terga furentis;
agnoscit gemitus et poenae parcere nescit,
quam subiit, dominique memor, quem verberat, odit.
adde, quod eunuchus nulla pietate movetur
nec generi natisve cavet. clementia cunctis
in similes, animosque ligant consortia damni;
iste nec eunuchis placidus.
And translated:
Nothing is so cruel as a man raised from lowly station to prosperity; he strikes everything, for he fears everything; he vents his rage on all, that all may deem he has the power. No beast so fearful as the rage of a slave let loose on free-born backs; their groans are familiar to him, and he cannot be sparing of punishment that he himself has undergone; remembering his own master he hates the man he lashes. Being a eunuch also he is moved by no natural affection and has no care for family or children. All are moved to pity by those whose circumstances are like their own; similitude of ills is a close bond. Yet he is kind not even to eunuchs.
Best Answer
Authorship
The byline to Jacula Prudentum reads "Selected by Mr. George Herbert, Late Orator of the University of Cambridge," and a note attached to an 1881 edition of the collected works of George Herbert says this:
If the English trail of the proverb ends at Herbert—either as author or translator from another language—it follows that Herbert is the source, in English at least. And indeed there is a related French proverb (discussed below), though it is not at all a literal counterpart of “Milk says to wine, Welcome friend,” but a possible source nonetheless. I haven’t been able to find any source in English that antedates Herbert.
The earliest instance of the expression that I've encountered in various book database searches is from Outlandish Proverbs, Selected by Mr. G.H. (1640):
Wikipedia confirms that "Mr. G.H." is George Herbert, and Outlandish Proverbs is evidently an earlier edition or version of Jacula Prudentum. Herbert died in 1633, so the proverb was clearly established by that date.
Early attempts to categorize or characterize the saying
William Hazlitt includes the phrase “Milk says to wine, Welcome friend” in his relatively systematic and well-organized English Proverbs and Proverbial Phrases (1869), but in discussing Jacula Prudentum, Hazlitt is not especially charitable:
W. Anderson O’Conor, “On Proverbs,” read on November 17, 1879, and included in Papers of the Manchester Literary Club (1880), sees this proverb and others in Herbert’s collection as representing a stage in the historical development of the proverb:
Though treating certain proverbs from Herbert’s collection as riddles may be defensible, doing so understates the extent to which they remain statements of folk wisdom. But what is the folk wisdom in “Milk says to wine, Welcome friend”?
Meaning of the proverb
My first thought was that milk was welcoming wine as a friend because both were agents of health in a human diet—rather as if one doctor were greeting a respected colleague. But that idea runs into some problems elsewhere in the land of proverbs.
The “milk says to wine” proverb also appears in [Robert] Southey’s Common Place Book, third series (1850). However, Southey also cites another saying from Herbert’s collection:
This saying is no less opaque than the “milk says to wine” proverb, especially with regard to the idea that a morning sun seldom ends well, unless it be that sunset inevitably comes to even the brightest morning (which hardly discredit the bright morning, in my opinion). The troubled future predicted for a "wine-bred child" at the very least suggests that the author of this proverb doesn’t consider milk and wine to be interchangeable in a child’s diet. The "Latin-bred woman," meanwhile, seems to refer to a woman who has received a sound and scholarly education—a potential source of friction in a society that prefers women to be intellectually submissive and unambitious. There is certainly no hint in the "wine-bred child" adage of a happy friendship between milk and wine.
Another perspective on the question arises in William Benham, Cassell’s Book of Quotations, Proverbs and Household Words, revised edition (1914), which repeats the “milk says to wine” proverb and then adds a cross reference to the following saying from John Ray, Compleat Collection of English Proverbs:
And Godfrey Baseley, A Country Compendium [combined snippets] repeats the French saying about wine and milk and then follows it with a new and very brief saying about the two:
The Benham and Baseley sayings suggest that the crux of the “milk says to wine” proverb may be that if you drink first milk and then wine, your stomach will accept the newcomer calmly; but if you reverse the order, your stomach will rebel.
Ray’s “wash milk from your liver” proverb—though it may just be a very loose translation of the same French proverb that yields the “milk on wine, poison” adage, seems to go a step further and suggest that something—whether wine or water or some other fluid—is necessary to wash milk from your liver, to avoid bad results from the milk. Ray himself, however, seems to have viewed the “wash milk from your liver” saying with considerable skepticism. In his comment beneath that proverb (which he identifies as “Gallic) from the 1678 edition of his book, he remarks:
A seventeenth-century critique of 'Milk must be washed from the liver'
Early English Books Online finds this interesting analysis of the above proverb, from James Primerose, Popular Errours.: Or the Errours of the People in Physick, First Written in Latine by the Learned Physitian James Primrose Doctor in Physick (1651), applying the principles of physiology (as they ere then known) to the case:
It thus seems that many people in England were accustomed to drinking wine soon after drinking milk on the supposition (or superstition) that this would prevent harmful effects that milk would otherwise have on the liver. Since this practice was evidently widespread in 1651, it may be that the latter saying was simply an alternative way of expressing the quasi-scientific justification for the practice—namely, that "Milk must be washed from the liver."
Still, it bears repeating that George Herbert had recorded "Milk says to wine, welcome friend" as a proverb no later than 1633, in a collection of proverbs published (posthumously) by 1640.