？
Would a real neurotic really call themselves neurotic?
Literal
Real [genitive-の] neurosis [genitive-の] guy [subject-が] neurosis [such-as-だなんて] self-proclaim [question-のかね]?
A rhetorical jab in casual register. ノイローゼ is a loanword from German 'Neurose' ('neurosis'), used in Japanese for neurotic mental states. Note the double loan: ノイローゼ (German) + 奴 (native 'guy, fellow,' slightly rough). だなんて combines だ (copula) + なんて (dismissive 'such as'), used here to topicalize the claim 'is neurotic' as something absurd to assert. 自称する ('to self-proclaim, to call oneself') is a compound: 自 (self) + 称 (name, call). のかね is the explanatory question form followed by a reflective ね — 'really, would they?' Used for rhetorical musing rather than genuine inquiry.