"What day is his trip?" was accepted. There are different ways to say the same thing across the board in all languages
Are they making a distinction between "what DAY (of the week)" and "what DATE (of the month)"? Sort of pushing it for English, I think.
Not really. There is a pretty clear distinction even in English.
If someone asked me "What date?" I'd automatically include the month in the answer. In my English, both ka lā ʻehia and ka Pōʻahia would usually be "When?"
For a more Hawaiian way of looking at (ka lā ʻehia) look at time segment [9:12] to [ 11:16] in the video link provided: