Yes, it is a Euphorbia, and yes, you can take a cutting and use that to start a new plant. Wear gloves and be sure not to touch the white sap that comes out under pressure when the skin is broken. You could cut here for example:
The mother plant will branch after the cut, and the cutting should develop roots within 2-3 months.
These plants are large trees in nature and while they do fine in containers, they seem to enjoy (or require) being cut down and restarted every few years.
