It's that sheet/blanket that they put on you so you don't get covered with hair. Is there a formal name for it?
I think you are thinking of a barber's cape. Sometimes you might also hear 'barber's gown'. See this commercial site Salon Lines, for instance.
My grandfather was a barber for 72 years, in Alabama and Florida. He called it a barber cloth (not the possessive barber's). This term can be found when ordering this product from Amazon.
I've been a barber for almost 40 yrs. The proper term for this article is a "chair cloth" or less commonly "hair cloth." The term "barber cape" or simply "cape" did not come about until the unisex trend became popular and the distinctions between barbering and cosmetology became blurred. In my shop it remains "chair cloth."
I would call that a cape:
A sleeveless outer garment fastened at the throat and worn hanging over the shoulders.
I see it being called a cape, both formally and informally, but it's really a bib. If you wear a cape backwards (so the slit is down the back), it's a bib. When kids are playing Superman, they wear their bibs backwards to make them... a cape!
I think adults just don't want to wear a bib, since that's a baby thing. Unless they're eating Lobster.
According to the 1911 barber book by A.B. Moler the answer would be chair cloth, a hair cloth was a strip of cloth used between the neck and the chair cloth as we now use sanek strips. You can call anything whatever you like but the 1911 book is the gold standard for barbering books and is still greatly quoted in every barbering text book that I have seen. My barbering instructor just died in nov of 2017 at the age of 91 with about 75 years under his belt and it was a 'chair cloth' to him.
I have heard the term "smock" used, without any additional qualifiers.
It is called as either apron/gown.