How many timesThey're a I seen a blind person wearing sunglasses though? Every single time I have ever seen a blind person. But glasses aren't practical to wear in a fight (won't stay on your head and will break, unlike a tied bandana), and Kenshi is designed with a feudal Japan flair, so sunglasses don't make sense. The bandana also calls attention to the fact that he is blind, because if he wasn't wearing one then players unfamiliar with his character wouldn't ever know he's blind. In terms of the game, it's a character design choice so that players will recognize that he's blind without reading his bio.
And yes, even if you can't see with your eyes, letting the sun scorch them will still hurt and lead to other problems.