As you may have already known, ye old Gallup test was developed by primitive people who believed that the reflection in the mirror is a trapped soul. That's why animals who (which?) don't have a soul cannot recognise themselves in the mirror. They can't have self-awareness. Several decades later a group of neuroscientists proposed a revolutionary hypothesis: that the mirror test actually measures if an animal recognises him- or her- (it-?) self in the mirror. In this paper they used fMRI to nicely show that "thinking of oneself" and "looking at oneself" activate different brain regions: https://www.ncbi.nlm.nih.gov/pubmed/12167262
In other words, recognising oneself in the mirror is something else than thinking about oneself. Thank you, Science!
PS, the child in the video is probably delayed.