You might also get good answers over at /r/academicbiblical, they strive to maintain the same high standards of moderation as this sub, and its focus is on the secular study and scholarly criticism of Jewish and Christian writings.

One narratively important thing is that God himself does physically record these commandments onto stone.

Many scholars have pointed out that throughout Genesis, God gradually becomes more absent. He walks with Adam and Eve, he chats and bargains (and maybe visits) Abraham, Jacob sees his messengers, he appears to Joseph in dreams, and so forth. Then, suddenly, in Exodus a God becomes more present again: he speaks to Moses through a bush, he give Moses detailed instructions and sends ten signs (Pharaoh was ready to give up but God “hardens his heart”, apparently to be able to keep demonstrating his ultimate power over all matter and life), and then, in a very interesting moment, rather than God coming down to earth, Moses is brought up to a high place. God’s presence is suddenly very, well, present. And the Ten Commandments play a big part of that—they’re quite literally written in stone, by the very “Finger of God”, according to both Exodus and Deuteronomy (interestingly, the Egyptian magician attribute the plagues they experience to the same “Finger of God”—I think those are the only three uses of the term in the Hebrew Bible).

Another thing scholars have pointed out is the covenantual language throughout Genesis and Exodus (and reoccurring elsewhere, most notably with the Davidic covenant and, in Christianity, the “new Covenant”). Many of great moments in the Hebrew Bible are sealed by such agreements/contracts/conditional promises/unconditional promises. They are, scholars frequently argue, written in the language of Ancient Near Eastern legally binding contracts. You can find a list here. The Mosaic Covenant, the agreement between God and the Jewish people to set them apart as a “kingdom of priests and a holy nation" (Exodus 19:6), is at its core agreed to on Mt Sinai and began with the Ten Commandments (it’s of course immediately partially reneged on with the Golden Calf and then reemphasized). The Wikipedia page says that “the Decalogue does not have any witness”, unusual for such contracts, but I remember the interpretation that all of Israel is gathered at Sinai/Horeb as both witness and participant in the contract. This contractual nature of the relationship is made unusually explicit as, for example, Sabbath is an explicit sign of this contract, this covenant.

31:12 The Lord said to Moses: 13 You yourself are to speak to the Israelites: “You shall keep my sabbaths, for this is a sign between me and you throughout your generations, given in order that you may know that I, the Lord, sanctify you. [...] 16 Therefore the Israelites shall keep the sabbath, observing the sabbath throughout their generations, as a perpetual covenant. 17 It is a sign forever between me and them people of Israel that in six days the Lord made heaven and earth, and on the seventh day he rested, and was refreshed.”

I think part of the confusion is that the word covenant is an unfamiliar one, and the scene would feel different if we substituted the more familiar “contract” or “agreement”. Some thing there are two tablets because, like any good contract, there are two copies—one for God, one for the Jewish people. The Finger of God which wrote the physical copies of the Ten Commandments has been interpreted as God’s very own signature on the two-sided contract wherein the Hebrews moral and loyal to God and God marks them out as a special, holy people. This moment is a major turning point in the Hebrew Bible, on par with the moment of Adam and Eve’s expulsion, Abraham’s call, Moses’ call, and the start of the Exodus. In some ways, it’s the culmination of all of the before (and then there will be a second culmination in David’s selection and the Davidic covenant).

One note is that the Biblical narrative itself calls these Ten Commandments. Actually, it calls them the Aseret haDevarim, the ten sayings/words/matters. The book of Deuteronomy is called in Hebrew “Devarim”, the same word, and this is usually translated as words as in the first line of that book, “These are the words that Moses spoke to all Israel beyond the Jordan”. So we know that there are ten of them, but different sects have counted the ten differently. There are perhaps 14 or 15 possible different things that could be Commandments/sayings, but we know there are ten because the text tells us there are ten in at least Exodus 34:28 and Deuteronomy 10:4.

These Ten Commandments, specifically the tablets that they’re written on (again, by the Finger of God), are the most powerful ritual object of Hebrews until the construction of the First Temple (they’re of course placed in the Holy of Holies of the First Temple where they seem to form a sort of footstool for a divine throne that is sort of the entire building). Even someone who touches the box carrying the Ten Commandments—with good intentions, as he’s jus trying to prevent it from falling over—is killed on the spot. You can only touch the long carrying poles which touch the rings that hold the poles in place which are attached to the box which hold these very sacred objects. There are accounts in Judges where the Ten Commandments Tablets (in the box held up by rings carried on poles) are taken into battle. When the Philistines capture them, it’s bad news for everyone. Those bits in Indiana Jones and the Raiders of the Lost Ark about the ark’s (box’s) immense power comes from the fact that the ark is carrying these very sacred objects.