Yes. Each ship is given 22 characters for a name and if your ship's name is exactly like someone else's, fear not! The game will not tell you that a name is taken. The same idea is applied to the ship's ID but you have 6 characters instead. As an example, I use KR-77X for my Anaconda's ID and Big Gun Hypothesis for the name. If you buy and spend some Frontier points, you can get a Name plate and ID plate for your ship so if someone gets close enough, they can see your ship's name and ID on the hull. Normally you just see it if you scan a ship.
Edit: Should also point out that if you decide to change anything in Livery, you are not charged Credits or Frontier Points for the alterations, so pick whatever you feel looks best with reckless abandon. Just mind your bank account. As you advance in Rank (Harmless, Aimless, and Penniless upto Elite), you unlock emblems to apply to your ship as well. You can also change your ship's name and ID anytime the Livery is available at a station for no cost.