In the newest filler episodes of Naruto (462 and 465), they show Hagoromo and Indra using some sort of lightning release that looks like the Chidori. So did Kakashi actually invent it? Or did he copy that technique just like all his other attacks and claim it as his own?
Here's the referendum from wiki:
The Chidori was created by Kakashi Hatake after he failed to apply his lightning-nature to the Rasengan
Maybe in the said episodes the ninja uses a variant of "Chidori". There are various altercations of Chidori that visually looks "similar", e.g. Raikiri.
So I guess what I am saying is, even though "that" particular form of chakra behaviour exists, it is possible (and canon) that Kakashi is in fact the one who invented Chidori.
It is very possible in Naruto-verse to invent a new jutsu even if they haven't been taught down the generations. But as canon goes, till this date, Kakashi is said to be the inventor of Chidori.
I mean, it was basically just added into the show because Studio Perriot wanted to show the whole reincarnation thing with Sasuke.
For an in-universe explanation though, it's probably just a case of Kakashi "inventing" something that he didn't even know was already a thing, hence why Indra uses the same handsigns and it makes the same noise.
Ever thought of a great idea, only to find out it's been done before? Same thing.