No, that would be food coloring. You could, though, if you wanted it to be red.
'Red cooked pork' isn't actually very red, not authentically. I guess someone at one point or another--probably in a restaurant here--got tired of being asked why the pork wasn't red and so they made it red. Like other non-Chinese 'Chinese food', it spread.
Cooking 'red' in China means to braise in a mixture of garlic, dark soy sauce, sugar, and rice wine. It is a common technique used by home cooks and has some regional variations--hot chilies, e.g., in areas where the cuisine is spicier, star anise in others--but the basic braising approach with these key ingredients is widespread and is also used for other meats, tofu, fish and seafoods. The mixture ends up looking shiny brown but can appear somewhat red, hence the name.