The original term "duck tape" appeared in the Oxford English Dictionary in 1899 - it was called that due to it being waterproof (c.f. "water off a ducks back") though it was a non-adhesive cotton tape, so different to the modern product. During WWII, Johnson & Johnson were given the job of producing the tape (for securing ammunition boxes) with self-adhesive properties and thus the modern duck tape was invented. After the war, the tape was mass produced and one of the common uses was to wrap air ducts, but the term "duct tape" didn't enter the dictionaries until 1965, although the term "Ductape" was trademarked in 1960 by Albert Arno Inc - their product was a flame-resistant tape that could withstand temperatures up to 204 °C. Note that modern "duck tape" is not heat resistant and it's generally considered dangerous to use on ducts as it can release poisonous fumes and catch fire.
However, "duct tape" is a term that is frequently used to refer to "duck tape", but is surely a misnomer as it's not suitable for use with air ducts (arguably not suitable for use with ducks either) and yet it is waterproof (like a duck).
Not necessarily as Duck Tape is a well known brand in the UK. Also, using it for ducts is prohibited in California and by building codes in other places (according to Wikipedia).
However, "duct tape" is a term that is frequently used to refer to "duck tape", but is surely a misnomer as it's not suitable for use with air ducts (arguably not suitable for use with ducks either) and yet it is waterproof (like a duck).