Does El Shaddai mean "the many-breasted God"?

No. El Shaddai means God Almighty, and it has nothing to do with a breasted God. No. El Shaddai means God Almighty. It has nothing whatever to do with a breasted God, many-breasted or otherwise. Scripture itself tells you what the name means, so we do not have to guess. In Exodus 6:3 God says to Moses that He appeared to Abraham, Isaac and Jacob as El Shaddai, and every major translation, from the Septuagint through the Vulgate's Deus omnipotens to the Douay-Rheims "Almighty God," renders it as almighty, all-powerful. Translators did not settle on that by accident. It is how the divine name functions in the patriarchal narratives, where God is announcing His power to make and keep promises that no creature could make good on: a child from a barren couple, a nation from one man, a land for a wanderer. El Shaddai is also one of the specific divine names, not a general one. Elohim is the general term. That is why Elohim can occasionally take in angels, judges or rulers, because it is a broader word that can be applied to God but is not restricted to Him. Adonai, Yahweh and El Shaddai are specific to God. So when Scripture reaches for El Shaddai it is naming the one true God in His omnipotence, not describing an attribute of nurture. Where does the "many-breasted" idea come from, then? It comes from an etymological guess that links Shaddai to the Hebrew word for breast, and it gets promoted today because it is useful to people who want to find a feminine or maternal deity inside the Hebrew Scriptures. That is the tail wagging the dog. You do not establish the meaning of a divine name by picking whichever consonantal root suits your theology and then reading the whole Bible through it. You establish it by how Scripture uses the name and how the Church has always understood it, and on both counts it means Almighty. Now, I want to be fair to what people are often reaching for here. There are a handful of passages, in Isaiah and a few other places, where God likens Himself to a mother caring for her child. Those passages are real and they are precious. But look at the grammar. The key word is "like." In Hebrew it is the kaph prefix, in Greek the hos. That little word does all the work, because it says the thing being described is comparable to a mother, not that it is a mother. God sometimes acts like a mother acts. That is about as close as Scripture ever comes to maternal language for God, and it is a simile, not a name and not a nature. The Almighty, by contrast, is a title the New Testament keeps right where the Old Testament had it. In Apocalypse 1:8 the Lord God says, "I am Alpha and Omega, the beginning and the end, who is and who was and who is to come, the Almighty," and the Greek word is pantokrator, the all-ruler. That is the same idea that El Shaddai carries in Genesis and Exodus, and John applies it to Christ as well as to the Father. Almighty is the thread. There is no maternal thread to follow. So when someone tells you the Hebrew "really" says many-breasted God, ask them for the evidence, and watch what happens. What you will usually get is an appeal to a disputed root, not to a single passage where the name is used that way. Hebrew is a compact language and people put a great deal more weight on individual roots than the language can carry. The Old Testament itself is simply not interested in the question they are asking. It is interested in telling you that the God who called Abraham has the power to do what He promised.