Looks more like a yucca to me.

The Yucca you have is Yucca elephantipes. Great tropical houseplant--it's not cold hardy so obviously it needs protection from hard frost. It loves full sun and let the soil dry slightly before rewatering. Yes, it could use a bigger container but then again, it'll grow even bigger. Use a well-draining mix--add sand. Yes, you can cut off the side shoots with a sharp, sterile knife and plant the shoots in a sandy soil. Being summer, leave the young pups outside with some protection from excessive sun/heat. It'll soon from roots. They will even root from cane cuttings without any leaves!--Easy! Good luck!--I love Yuccas (native or otherwise). Yours is from southern Mexico I believe.

That is one huge Yucca Elephantipes! My only concern with the small container is that it's vulnerable to blowing over in strong winds and the head breaking off. A heavier container might help to avoid that, especially during strong storms.

NOT A PALM

I agree with what everyone is saying. And dont worry about repotting, these yuccas are BULLETPROOF! As long as they dont get frozen, they cant be gotten rid of! I dug some up a few years ago and kept it in my unheated garage (completely bareroot all winter with temperatures only in the 40s in the dark and cold garage) and I was able to get cuttings from it during the spring and they rooted within a week! They are super easy plants they can be overwatered, underwatered. and everything inbetween. I would recommend giving them the best care possible, but it seems like you are doing a great job! Just repot them.