The slogan "Sometimes you feel like a nut, sometimes you don't" is correct since an almond is a nut. A nut is defined (among other things) as being the seed of a fruit.
The almond is frequently, but incorrectly, said to be a member of the peach family.
The almond is not a member of the peach family, but rather it is a member of the Rosaceae (Rose) family. The peach also is a member of the Rosaceae (Rose) family. Thus the almond and peach are in the same family but neither is in the other's family, as this term (Family) is commonly and taxonomically understood.
Some other well known consumables that are in the same family (Rosaceae):
Rosaceae Prunus Dulcis - Almond
Rosaceae Prunus Persica - Peach
Rosaceae Prunus Domestica - Plum
Rosaceae Prunus Cerasus Marasca - Maraschino Cherry
Rosaceae Prunus Armeniaca - Apricot
Rosaceae Prunus Spinosa - Sloe
https://en.wikipedia.org/wiki/Almond_Joy#Trivia