There’s possibly something there though, even if the steak tartar version is incorrect. Bret Deveraux claims the aim was to preserve meat into a kind of jerky: “On the move, meat could be placed between the rider’s saddle and the horse’s back – the frequent compression of riding, combined with the salinity of the horse’s sweat would produce a dried, salted jerky that would keep for a very long time.”[1]
His stuff is very well sourced, but also he’s self-admittedly talking outside his area of expertise there. So who knows.
[1] https://acoup.blog/2020/12/11/collections-that-dothraki-hord...
I had always assumed they had the same etymology. It didn't help that in Spanish all of these words maintain the 'r' between the 'a' and 't': tártaro, ra, Tártaro, Tartaria.
Here there are 3 different etymologies: https://dle.rae.es/t%C3%A1rtaro and here there's another one: https://fr.wikipedia.org/wiki/Sauce_tartare
Although I don't know if the Wikipedia article should be trusted, since the English version one (Tartar sauce) states: "The word Tartar is a Turkic word that is the name of Tatar people."
Seems like a western adaptation of tatziki or tarator sauce, plus some orientalist confusion over who to call “tartars”. Interesting stuff!