Henry VIII is one of my favorite examples for the problems of names and ordinals.
Wikipedia says (correctly, I think) that Henry originally used the style "Henry the Eighth, by the Grace of God, King of England, France and Lord of Ireland". That's as close as we can get to a "name".
Ignoring the Display Name, he could be:
First Name: Henry VIII, Last Name: King of England. Nice, but King of England wasn't really his last name, and what about France? He didn't rule France, but it was part of his name / style.
First Name: Henry, Last Name: Tudor, Suffix: VIII. The problem is that he wasn't the 8th Henry Tudor. Also, by custom sovereigns have no last name. He belonged to the Tudor dynasty, but his last name was not Tudor.
First Name: Henry, Last Name: of England, Suffix: VIII. This is the best approximation in my opinion, but still not perfect. First, the ordinal wasn't part of his birth name. And, even if we ignore the birth name convention for titled people, we leave out the "and France" that was arguably part of his birth name.
There are many variations of the above examples, but this captures my main points.
As Victar said, it's a balancing act. Trying to find the solution that is only minimally wrong, and is easy for people to understand.