It's slang for: That's not the way I behave/act.

It's slang. It could mean any of these: "I don't like doing things that way." "That's not the way I dance." "That's not the way I have sex." (If I said "I don't get down like that", this is what I would mean) It depends on the context!

'Get down' is slang for 1) to party 2) dance 3) have sex The exact meaning here would need the context. Probably from "Let's get down to it" = Let's get started