Chai is how you say tea in lots of central Asian languages. So Chai Tea is Tea Tea.
DH's family makes the most revolting milk chai ever. It is boiled up with milk and it has chunky bits in it and a skin on top. . If you ever visit Afghans and they offer you tea say no. Either it's black or green and it's been brewing until all the flavour is gone and nothing is left except bitterness or it's the vomitous white stuff. I feel a bit queasy now.