Why a genitive here?

Diogenes Laertius preserves a letter in which Epicurus explains how we form the concept of time.
After doing so, Epicurus warns that we should stop and not keep going:

οὔτε ἄλλο τι κατ’ αὐτοῦ κατηγορητέον
Nor should we predicate anything else about this

ὡς τὴν αὐτὴν οὐσίαν ἔχοντος τῷ ἰδιώματι τούτῳ
as having the same essence as this distinctive property

Why is ἔχοντος genitive? I expected it be accusative, to match ἄλλο τι.

Is it matching αὐτοῦ, the object of κατ’?

If so, why? If not, is my reading (which is like others’) wrong?

I think that ἔχοντος matches αὐτοῦ:

Your translation seems fine. ἄλλο τι is the whole ὡς τὴν αὐτὴν οὐσίαν ἔχοντος τῷ ἰδιώματι τούτῳ