Its name is likely based on the "Stuck on You" toy phenomenon, which had its roots based on the comic strip character Garfield.
As you do not have to vanquish ten players in one session, Stuck on You is relatively easy to get if you use the Scientist a lot. Either stick the Sticky Explody Ball to plants that have low health, or lay them down to defend teleporters in Gardens & Graveyards. Try to throw a Sticky Explody Ball at a plant before you start a duel with them. If it sticks on the plant, shoot the plant a few times and let the Sticky Explody Ball do the rest. This is easier to do with the Sticky Cheetah Ball because it deals 75 damage to plants as opposed to 50 with the normal ability.