Answer: Bottle Palm is also called Ponytail Palm. It is known botanically as Beaucainea recurvata or Nolina recurvata. (Silly botanists, can't make up their minds!) The plant usually has only one stem until it reaches 3 feet tall, except if the main stem is damaged. The plant is propagated through seeds or suckers. I suspect the shoots you're finding are actually suckers from the roots because the main stem was damaged. I think I'd allow them to mature a bit and then carefully separate them, taking some roots with each new shoot. Give them 2-3 months to develop sturdy root systems before separating them from the parent rootmass. Instead of a single plant, you'll have several.
Best wishes with your new plants!
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