These are the verses:

Over the desert wild and free,
Rides the bold Sheik of Araby,
His Arab band at his command,
Follow his love's caravan.

Under the shadow of the palms,
He sings to call her to his arms.

When stars are fading in the dawn,
Over the desert they'll be gone.
His captured bride close by his side,
Swift as the wind they will ride.

Proudly he scorns her smile or tear,
Soon he will conquer love by fear

"He," 'his", 'They'll", yup definitely in third-person these verses. Now if the whole song were like the chorus you'd have a point. But do you see what I mean?
The verses are sung by a third-person narrator who describes what the Sheik is saying/doing. The perspective shifts to the Sheik in the chorus.