This true,Head pets are like one of the most common tropes in Japanese anime/manga. It usually denoted a nurturing, caring, or protective personality.
In Sazh’s case, he’s a strong-willed dad who’s son is his driving motivation.
The baby Chocobo is visual shorthand for Sazh’s goal of finding/protecting his son after the death of his wife.
And not wanting people to touch or disrespect our hair is a common trope in black culture. It's especially common for foreign people to violate us in this way because we have a unique texture that they are not familiar with. When this rule is not followed, it is usually to denoted to a lack of connection with your blackness or c00ning.
The baby Chocobo using his hair for a birdsnest is visual shorthand for the the studios lack of diversity behind the scene. And sign that this is probably going reflected in other aspects of his characters writing.