Fun fact: mad hatters were a real thing. The mercury used to process felt caused mad hatter syndrome, characterised by "behavioral changes such as irritability, low self-confidence, depression, apathy, shyness and timidity, and in some extreme cases with prolonged exposure to mercury vapors, by delirium, personality changes and memory loss, difficulty with social interactions. Associated physical problems may include a decrease in physical strength, headaches, general pain, and tremors, as well as an irregular heartbeat." (Wikipedia)
I doubt it, because Data was explicitly dressed as Sherlock Holmes when he was wearing this hat. And really, the very fact that you called it "a Sherlock hat" is indication enough that he's the only answer really needed.
Data could be considered the fictional character (who also happens to be wearing a hat that is iconically associated with the fictional character Sherlock).
On the other hand, Data could be considered the "actor" playing the fictional character Sherlock. Regardless, the hat is unmistakably an icon of Sherlock - I'm not making a serious case for accepting Data as an answer. But still....
i couldn't get the person in the trump supporter hat, the pirate that i thought was a girl or worzel gummidge that turned out to be a wizard.
On the other hand, Data could be considered the "actor" playing the fictional character Sherlock. Regardless, the hat is unmistakably an icon of Sherlock - I'm not making a serious case for accepting Data as an answer. But still....
*Sighs*