Which is better for you protein shakes or chocolate milk

Chocolate milk can have a lot of added sugar, so if that is something you watch, you might want to go with a protein shake.

edited to say that I get cytosport whey protein, low sugar, 27g protein, decent BCAA profile, and cheap (6lbs for around $50 at costco).

I would go with Chocolate milk for the following reasons:
1. Far cheaper per unit of volume and unit of usable nutrients. Protein shakes are expensive.
2. It is what it claims to be (assuming you get a good brand without a bunch of artificial sweeteners). Many protein shakes and meal-replacement shakes are packed full of dubiously-effective 'super-ingredients' that are often not at all effective or not useable by the body - they're just there so the company can claim 'the highest levels of x!!' without any evidence that any level of 'x' at all is useful, and charge you more because of it. Chocolate milk is milk and chocolate. Protein, fat, sugar. No supplementary algal culture, magic ionic alfalfa or proprietary blend of pseudo-scientifically named whatever.
3. Less processed - if you value whole food, supporting primary producers and avoiding highly processed products, chocolate milk wins hands down.
4. Milk contains plenty of protein. Western diets are grossly oversupplied with protein as it is, and the faddish desperation for very high protein diets is unproven to be of value for normal, healthy-weight or overweight adults. Additional protein after a workout most likely provides very little benefit to someone on a restricted calorie diet where muscle gain is likely to be minimal.