It is an aloe, but it's hard to say which one. I typically leave the dry leaves on the stem unless they let go on their own, but if you like you can trim them back to the stem with a pair of nail scissors. Maybe give the stubs near the base of the plant a tug to see if the last remaining part is still firmly attached.
I have the sense that your aloe has the potential to be a much bigger plant, but has been underwatered. Or maybe the roots are starting to give way. Does the plant still hold tight to the soil if you tug gently on the base of the stem? What is the recent watering history?
More about the genus here.
The Aloes Database
