The love potion will allow your mating-age adult to mate-by-connection with another mating-age adult (of the opposite gender) even if it's the first time they've met each other.
There is no permanent clone / twin. However there is a password ( 32479 91490 ) from the Nintendo DS game "Tamagotchi Corner Shop", which when entered into a Connection/Connexion V3 will give you an item labeled "!!", which can be used by an adult Tamagotchi character to summon a clone of itself. Your Tamagotchi character and its clone will then dance together for a few seconds before the clone disappears.