Something similar was discussed at the university on a pair of "psychology of journalism". If the journalist, after the answer of his interlocutor in an interview, repeats the words (usually a tone lower, or even in a whisper), this is so that the interlocutor continues his thought. That is, he sees that the journalist has repeated the last words and does not ask anything, most likely he will continue to speak on the same topic.
In connection with this question, one literary example comes to mind. Mikhail Gendelev, Great Russian Journey, chapters 22 and 23 (http://heblit.bravepages.com/mg/mggrt.html#_hd126)
“- And in general, - said the mother, - what are you, did you come to our country to behave like a rogue?
Echolalia is not a "syndrome", but a symptom of a number of mental disorders (I hope that psychiatrists will come to this question and supplement literary examples with purely scientific ones). It happens that this is not associated with disorders, in children, for example, it may just be one of the stages of the formation of speech.