Early instances of the saying in English
A 1634 translation of Matheo Aleman, The Rogue: Or, The Life of Guzman de Alfarache, third edition (1634) offers an interesting gloss on the reasoning behind this old saying:
Men ought to make the same choice of their friends, as they doe of good bookes. For their happinesse consists not neither in the greatnesse of their number, nor the curiousnesse of their binding, but rather to have a few, and those good and approved Authors. For oftentimes, it falleth out, that many friends are rather a hindrance, than a helpe to true friendship: For he is a friend to none, that is a friend to all. Nor are we to choose a friend for our entertainment only,and merrily to spend the time withall, but to make choise of such a one, as may benefit both our soule, and our bodie: one that without respect of humae interest, will advise him to observe the divine precepts. Not only to represent them unto him, but to speake to the purpose, to admonish him to the observation of them, and to instruct him in the true knowledge of them. And if he be called a true friend who, meerely out of friendship, tells his friend the naked truth without masking or disguising it, not a s athird person, but as to one, that is properly his owne, and as freely as if he should treat of any thing belonging to himself, or that he himselfe would desire, his friend should say the same to him, on the same occasion, of which sort of friends, that carrie that sincerity and plainnesse with them, few are now a-dayes to bee found, in whom a man may have that assured confidence, and entire satisfaction.
Another perspective, this one with an overtly religious edge, appears in Samuel Crook, Ta Diapheronta, or, Divine Characters (1658):
But he that is a friend to all men, is a friend to no man, and least of all to himself. For he must promise so much, that he cannot performe withall: and so breaking promise with some, he is trusted at length by none. Therefore, as he who boasted he had no enemy, was well reproved by Chilon, when he asked him, if he had any friend. So he is farre from a capacity to be truly befriended of any, who is seemingly a friend to all; but really a friend to none, and, encreaseth enemies by pretending friendship.
On the contrary, The true Christian is a friend to none that are enemies to God.
Interestingly Joseph Symonds, Sight and Faith, or Meditations on 2 Cor. 5.7 (1651) makes the universal love of Jesus particular to each person as if to avoid this criticism:
Christ is so a Friend to every one, as if he were a Friend to none besides; and hence it is, that they not onely said, Our Lord, and our God, but my Lord, and my God.
William Seward, Anecdotes of Some Distinguished Persons, Chiefly of the Present and the Preceding Two Centuries, second edition, volume 3 (1795) has this:
Guicciardini [1483–1540] again observes, "As he that is a friend to all is a true friend to none, so that which has many heads, has in reality no head at all. A Multitude is this many-headed monster, which hath neither a head for brains, and most assuredly no brains for government. ..."
Clearly, "A friend to all is a friend to none" has been an adage in English for several centuries.
Aristotle on friendship
Although Aristotle never wrote (in Greek) the terse wording "A friend to all is a friend to none," he seems to have agreed with the sentiment. He analyzes the nature and limits of friendship in Book 9, chapter 10, pages 1170–1171 of the Nicomachean Ethics. First he distinguishes between "friends made with a view to utility," "friends made with a view to pleasure," and "good friends." Then he focuses on the natural constraints on the third category:
So for friends, too, there is a fixed number—perhaps the largest number with whom one can live together (for that, we found, is thought to be very characteristic of friendship); and that one cannot live with many people and divide oneself up among them is plain. Further, they too must be friends of one another, if they are all to spend their days together; and it is a hard business for this condition to be fulfilled with a large number. It is found difficult, too, to rejoice and to grieve in an intimate way with many people, for it may likely happen that one has at once to be happy with one friend and to mourn with another. Presumably, then, it is well not to seek to have as many friends as possible, but as many as are enough for the purpose of living together; for it would seem actually impossible to be a great friend to many people. This is why one cannot love several people; love is ideally a sort of excess of friendship, and that can only be felt towards one person; therefore great friendship, too, can only be felt towards a few people. This seems to be confirmed in practice; for we do not find many people who are friends in a comradely way of friendship, and the famous friendships of this sort are always between two people. Those who have many friends and mix intimately with them all are thought to be no one's friend, except in the way proper to fellow-citizens, and such people are also called obsequious. In the way proper to fellow-citizens, indeed, it is possible to be the friend of many and yet not be obsequious but a genuinely good man; but one cannot have with many people the friendship based on virtue and on the character of our friends themselves, and we must be content if we find even a few such.
[Translation from the Oxford University Press edition, W.D. Ross, editor.]
The words boldfaced above may well be the ultimate source of the expression, "A friend to all is a friend to none"; but it is important to bear in mind that Aristotle is concerned in this discussion with intimate friendship, not merely the presumptive good will of fellow citizens. There is no objective constraint on the human condition that prevents one from treating everyone else politely and pleasantly. But Aristotle argues that there are natural constraints that prevent one from being everyone else's intimate friend. Being a close friend entails certain obligations such as availability, sympathy, shared ethical standards, and shared emotions when the friend feels elation or grief.
That's the context for Aristotle's analysis of the practical limitations governing how many friends a person should (and indeed can) have. His analysis also seems consistent with the reasoning underlying the assertions of Aleman, Crook, and Guicciardini above.