Tools used to spread oil for crêpes
Having now been able to watch the videos, they do seem like what I've seen. The first isn't very clear but looks like cloth rather than paper towel to me. The second looks a bit more manufactured, but again familiar. It's a (presumably cotton) cloth, possibly cheesecloth, rolled, and kept rolled with a crimped handle or foil. A cotton or linen tea towel (AKA dish towel) would work too. The handle also means less oil getting on the cook's hand. The only slightly odd thing is the foil at the bottom.
Cheesecloth is primarily meant for use in contact with food, and tea towels for contact with cooking equipment. Lint isn't an issue after a few washes.
The reason they don't burn is the very brief contact time, combined with a lot of the heat going into the oil; they're saturated by being kept in a dish of oil. This is a quick way to get a sufficiently even layer. If you look closely you may see that the spreader is used in a brushing motion so contact between any one part and the griddle or pan isn't continuous, meaning even less time to scorch. And after all, a cotton cloth can be used as an oven cloth.