He said he woudl never charge at his opponent like that because he woudlnt be able to avoid a counter attack since a forward charge is so easily counterable. However he admitted that now that Sasuke gained the same speed as him, that having Sharingan gives him a very important edge.
However based on the fact that Kakashi had Chidori before he attained Sharingan, I will safely say that you do NOT require Sharingan in order to use Chidori. And that Sharingan is not at all required (unless you wanna say that Kakashi some how knew that Obito would die and give him an eyeball lmao). Thus I'd say Kakashi wasnt refering to Sasuke being like him personality wise nor talent wise, but affinity wise seems to be MUCH MORE likely given the jutsu that he taught him.