The ability to gather Seaweed Stalks and plant them elsewhere was added to the Shipwrecked DLC more than a year after its official completion in the Home Sea Home update, after being tested in a beta branch for several months.
While Seaweed Plants can grow naturally in Coral Reefs, they cannot be planted there by the player.
Characters do not have specific examination quotes for the Seaweed Stalk and will always say "It's a... thing." when examining it.