The "generation" part is only part (one syllable) of the given name and doesn't constitute a separate name. The "yong" in my name is "generation name", but no Korean will call me "Yong" any more than an English speaker would call Richard "Chard".
Because most Korean names are neatly split into three characters (each character is a syllable in Korean), many people also write their names in three space-delimited words in Latin alphabet. And then westerners get confused and end up with patterns like "Gil D. Hong". (I'm not blaming them; you can't expect everyone to understand all the world's naming systems.)
Just to push back on this a bit, the fact that the "generation name" part of the name carries some additional semantic information in some ways makes it even more of a name than a "middle name" that is somewhat popular in the West. In a sense, my name is my full name (first, middle and last), but it is split into three parts. The last one has some meaning (it's a family name), the first two are given names, and the cultural default here is that I would go by my first name, and if a conflict is found, I'd go by the first + last, and if a conflict still exists, you'd default to first + middle name or first + middle initial. I don't think it's that unreasonable to consider generation names to be "compound names", consisting of two sub-names, even if the cultural default is to always use the full compound and not either of the parts.
I also think that "Richard"->Chard is a disingenuous choice, because people named Richard often go by Rich. People named Andrew very frequently go by either Andy or Drew, similarly people named Alexander often go by either Alex or Xander. Of course, there's no semantic meaning associated with the component parts of those names anyway, so it's not like you can infer something from the fact that someone is named Alfred and someone else is named Albert, and either one might go by "Al".
Well, OK maybe Richard was a poor example. What I'm saying is: some Richards go by Rich, but few goes by Chard, and certainly nobody interprets Richard as a combination of two name components "Ri + Chard" (or "Rich + Ard"). The probability a typical Korean would consider their given name as a combination of two parts is probably higher than that of Richards, but not much higher.
Maybe a better example is Anderson, which historically meant "Anders's son", but few living Andersons would consider "Anders" an acceptable way of writing their family name.
Because most Korean names are neatly split into three characters (each character is a syllable in Korean), many people also write their names in three space-delimited words in Latin alphabet. And then westerners get confused and end up with patterns like "Gil D. Hong". (I'm not blaming them; you can't expect everyone to understand all the world's naming systems.)