Well it's both but known as La Flor de Noche Buena here in Mexico (Christmas flower). Mine was one of three plants in a pot that I bought for the Cissus vine. They were so entangled I planted them together as is. The third plant is long gone but the Noche Buena grew to about 2.5 feet last year and was very pretty but didn't look like the Poinsettias you see up north. They are considered annuals up north because they can't take the cold but are native to Mexico and are actually perennials.
So for some reason last winter the plant looked old and tired and nearly dried up - so I cut it back to nothing. Very surprised a few months ago to see a few popping up near the original and now coming up as much as 10 feet away. So many that I have been digging them up and putting them in pots. So the question is are these starting from seed as there is no sign of runners off the original. They have individual root systems. Noche Buena is classified as a shrub or tree and can grow to 10 feet so we'll see what happens to some of these this year with better care.
The Poinsettia got it's US name from Joel Poinsett who was the first ambassador to Mexico and he introduced it up north. Lots of legends in Mexico from Early Christian times back to the Aztecs which are fun to read but pretty typical legends.
What you´d expect to see up north around Christmas
Mine in the garden
A few I´ve put in pots