I'm not sure of the name of this plant. It doesn't grow too quickly, maybe one or two inches each year.
Is the growth in the bottom picture a root? Can I clone this plant from that?
(Click on the pictures to see closer views.)
This is definitively a Yucca elephantipes. Not Dracaena.
It is indeed an air root coming out of it. However, this plant is one of the easiest to propagate, you can just cut of the top of the stem or your rooted shoot and put it in the ground. No hormones or sterile soil needed. Also without these air roots it will work out fine, just make the soil not too moist (it will cause rot). I have done this many times with my own Yuccas. You can even cut the main stem in pieces (of ~10 cm) and put them in the ground, they will all grow into new plants. From my own experience, best time to propagate is in spring time, not now during the winter.
This plant is very easy to keep and maintain, the only tricky part is watering it too much. The roots will then rot. Furthermore, try to dust the leaves off once in a while. You can do that with a damp cloth.
To clone this plant using that 'adventitious root' with photosynthesizing leaves will most certainly work. Are you wanting to grow another plant or would you like to make sure this one survives? Your pictures are very fuzzy. Is the mother plant healthy?
I am thinking that Bamboo's ID is closer than my original,marginata. Massangeana is closer if not perfect.
And I am now just seeing that this shoot is at the end of the live trunk. I would cut the entire piece of the end of this trunk from which this baby plant is sprouting, dip in rooting powder and place in firmed potting soil to grow to maturity. This is already a CLONE. It has the differentiated cells of roots and photosynthesizing material...all ready to go. Easy, you can't go wrong. Sorry, I didn't notice this start was at the end of a main trunk. Corn plant does this regularly. If you chopped off a main branch this would happen right away. Just cutting the 1 or 2 inches off the end of this branch to remove this start to be grown as an individual will cause more of these starts further down this branch/trunk as well as enervating other nodes on nearby branches.
Plant in firmed potting soil in a 4" pot with hole at bottom...Keep the top 1/4 inch of potting soil moist not wet, no fertilizer yet. Cover for a week with saran wrap plastic. When you see roots coming out of the bottom in a few months, I would up pot to a 6" pot. This plant likes being root bound and it should be happy in this 6" for months if enough light. Keep it out of direct sunlight or it will fry. Open the plastic to exchange the air. Light 18 hours with 6 hours of darkness. Best done with real grow lights. If you don't have enough light this plant will not be able to make enough food for energy for repairs, growth. This is a great project that you should easy have success with...Bamboo's ID is correct, I humbly think!