Can you say about someone that he wears his demeanor?
This is the specific sentence: "I was wearing my usual peevish demeanor"
It's grammatically correct.
"Wear" can mean:
So, wearing a characteristic of personality is perfectly fine.
"Showing" or "displaying" would be more natural than "wearing". Imagine a peacock's tail. It wears it all the time, but doesn't necessarily display it.