It depends on the time of year. In the summer heat, they use a thin linen material. In the cooler months, they use towelling material. What you have to bear in mind is that they have to live as pilgrims in fairly austere surroundings and mainly sleep in tents. If they get cold, there is no running to a heated area or wearing a jacket. It is like shedding your worldly possessions.
It is a Muslim tradition not just in the Middle East. But other than in Mecca (where they are all heading to), you only see them at airports around the world making their way to Saudi Arabia. Strange time to year to do an Umrah (the name of a mini-Haj which can be done at any time of year) since it is currently the Muslim fasting month as well so it is physically very demanding. I trust the individual was not eating or drinking in the lounge. I always think it is strange that while they have to live in austere surroundings in Mecca, flying in First or Business is not an issue.
If you want to see mass crowds in similar robes, head to airports around early December since the time for the Haj is just before Christmas I think.