Oh look, it's Redguard headcanon time again
Now I doubt I'm the first person to make this connection but I'm writing it anyway because I'm not sure I've actually read any posts about it here, so here goes.
Now, we don't know tons about how the ancient Sword Singers actually go about singing swords into existence, but we know that master Ansei can fight without conventional weapons, some can do it naturally but most take years to learn the techniques, and that there are other abilities besides forming a Shehai such as the Ephemeral Feint, suggesting it's a general style of magic rather than a specific ability.
Anyone else see similarities with anything else, perhaps another kind of power centred around one's voice, something that some can do innately but often takes years of rigorous training, something that can allow its masters to fight great battles with nothing but the strength of their souls?
Now I'm not saying Sword Singing and Thu'um are exactly the same power, they certainly have different spiritual origins, but I definitely think the actual magic is two variations of the same idea even if they came about through different beliefs.
However (and this is where I get a bit more tenuous, feel free to jump off the headcanon train now), there are some spiritual similarities here.
The Thu'um originally came from the dragons, great serpentine beings believed to have some godly aspect. It could be coincidence that the traditional Yokudan gods are also often serpentine, but bear with me. The Thu'um is stated in one book to be similar to the kiai shout of the Tsaesci of Akavir, serpentine warriors who are claimed by some accounts to need no shields or armour and fight only with their powerful blades.
I know we have no explicit evidence of Akaviri contact with man or mer until long after the both Sword Singing and Thu'um came into existence, but Varieties Of Faith explicitly states Yokudan god Diagna as originating in the "Twenty Seven Snake Folk Slaughter", and I only know of one type of snake folk.
So, trying to sum things up:
I think the Tsaesci learned kiai as a form of Shout from the Dragons (who some say come from Akavir), and the Yokudans learned it from them then adopted it as Sword Singing. Thu'um probably came directly from the Dragons to the Nords, making it a cousin of Sword Singing, but part of me thinks it's feasible that the Autumn Of Snakes, in which ancient Nedic towns were apparently invaded and devoured by great serpents (remember how the Tsaesci supposedly ate the men of Akavir?), was actually a real event and not simply widely-described apocrypha as one condescending Altmer book claims, considering the event supposedly took place in modern Hammerfell, which is obviously close enough for Tsaesci to reach from Yokuda, so through that it's feasible the Thu'um was adapted from kiai by the ancient Nedes, who intermingled with the Nords who came over later from Atmora and spread the practice to Skyrim, where it strengthened Dragon worship to become dominant among the Atmoran animal pantheon.