Has Menthyl nicotinate anything to do with nicotine?

NO, there is no functional relationship between Menthyl nicotinate and Nicotine. Niacin or Vitamin B3 was originally given the name of Nicotinic acid by Austrian chemist Hugo Weidel in 1873. Having obtained it by oxidizing Nicotine he called it “Nicotinic acid”.
The discovery that Niacin was a vitamin was made in 1937 by the American chemist Conrad Elvehjem. A few years later, in 1942, the American Medical Association gave this substance a more popular name less misleading than Nicotinic acid, to prevent illogical, unfounded associations between the harmful effects of Nicotine and the therapeutic ones of vitamins. Its new name became NI AC IN, from the first two letters of NIcotinic ACid, and the last two letters of vitamIN.