Mesh analysis is a way to solve circuits USING Kirchoff's loop law. Technically, you are just using the loop law but the whole procedure that you do in this video is called mesh analysis because its commonly used method
It looks like he was using the node he was entering to determine the sign. Because not only was the voltage switched when passing through the battery, but the voltage rose through the resistors. It's ASC (as opposed to the more common PSC), and it came out of left field, but I suppose if applied consistently, it's fine. But I agree, it's not my go to.
Simply because he used the sign convention that if you go from - to +, which you encounter the negative sign first, you treat it as negative. And if you go from + to -, like on resistors, you treat it as positive.
