The German states were only unified in 1871, 17 years before Frederick ascended to the throne. He would have been Frederick I as leader of the unified German states (Germany), as he was the first of his name to rule over unification, or Frederick IV as leader of the Holy Roman Empire, as he would have been the fourth Frederick to call himself emperor of the HRE should the unified states have been viewed as a continuation of that legacy.

However, in addition the various legacies, these various titles came with differing responsibilities and legal complications. As Frederick had previously held the title of His Imperial and Royal Highness Crown Prince of Germany, Crown Prince of Prussia, on his ascent to the throne his title became His Imperial and Royal Majesty Emporer of Germany, King of Prussia. There had previously been two Fredericks with the King of Prussia title: Frederick I (the first King in Prussia, crowned 1701) and Frederick II (Frederick the Great). It was on the advice of his counselors that he chose to keep the title King of Prussia, making him Frederick III.

That's an interesting question! I assume you mean the Frederick III reigning briefly in the 19th century.

He basically had three options:

• He could go with Frederick I, emphasizing his role as the emperor of a new, Prussian-led Germany as the first of his name.
• He could go with Frederick IV, emphasizing a continuity with the HRE, Frederick III having been emperor in the 15th century.
• He could go with Frederick III, emphasizing his Prussian heritage, Frederick II having ruled Prussia in the 18th century.

I'm missing the primary sources to back this up and would have to dig a bit, but as far as I'm aware, the reasoning was based on the constitution, with a dash of Prussian pride. The constitution said "The president of the [German] confederation shall be the King of Prussia, who holds the title of German Emperor". So since emperor was just a title, and was awarded due to the virtue of being Prussian king, one line of argument (that prevailed) was that the numbering should follow the Prussian line.

As an aside, neither his father nor his son had to deal with this question, because there hadn't been any William before as emperor or Prussian king.