"chai tea" is a kind of tea
Chai is the Hindi word for tea. So "chai tea" literally translates to "tea tea". You would not order a "naan bread" or a "soup of soup". The phrase is a marketing artefact, popularised by American coffee chains in the early 2000s.
What people usually mean by "chai tea" is masala chai, spiced milk tea. Just call it chai or masala chai. It's shorter and accurate.