Mitanni (or Mittani) had an Indo-European superstrate, and many names may be explained by Indo-Aryan languages, like Prītāśva "whose horse is dear" (Mayrhofer II 182). We can see the Indo-European name of the horse “aśva”, Lat. equus, etc.
It seems that Mitanni “ašušanni”, in spite of its similitude with Indo-European, is actually a Semitic word. We cannot se in “ašu-“ Indo-Aryan “aśv-”, but we could think to a Volksetymologie for its *a-.
Answer to Van Hooijdonk (Dienekes’ Anthropology Blog)
I've read that some Semitic and Egyptian words for horse are loans from some IE languages
such as Hebrew sus, Akkadian sisu, Ugaritic sisw, Aramaic sisya, Egyptian ssm, I think we can also connect Arabic šušan šuš (quick young female camel) and Arabic šawšan (camel servant). All those words stem perhaps from the word such as Mitanni Aryan ašušanni wich means horse trainer however on light of this discovery it could be that it's from one(s) of the Semitic languages that IE languages borrowed this word because:
1/this word shows very aberrant reflexions on each one of the IE languages it is attested on so that we cannot even provide a secure form of the proto IE word for horse
2/this word seems to lack an internal semantico-phonetical motivation within IE, while it looks very well rooted and explainable on Semitic motivation
Indeed the root "šwš" means to be quick and hurry in Semitic
3/this word is attested in Semitic and Egyptian as early as script begins
But of course those are speculations that need to be confirmed by archeologists and linguists.