I edited my Chinese names for OCs tutorial to add a section about generation names. Here is just the new section:
Starting in relatively recent history (not really a big thing until Song dynasty) and continuing, moreso outside of mainland China, to the modern day, there is something called a generation name component to a name. This means that of a name’s two characters, one of the characters is shared with every other paternal line relative of that person’s generation; historically, usually only boys get a generation name and girls don’t. (Chinese history, banging on pots and pans: DAUGHTERS AIN’T SHIT AND DON’T FORGET IT!) “Generation” here means everyone who is equidistant descendant from some past ancestor, not necessarily that they are exactly the same age. For example, all of ancestor’s X’s sons share the character 一 in their names, his grandsons all have the character 二,great-grandsons 三, great-great-grandsons 四 (I just used numbers because I’m lazy). By the time you get to great-great-grandson, you might have some that are forty years old and some that are babies (because of how old their fathers were when they were conceived), but they are still the same generation.
In some clans, this tradition goes so far as to have something called a name poem, where the generations cycle, character by character, through a poem that was specifically written for this purpose and which is generally about how their clan is super rad.
If you want to riff off of this idea and have siblings or paternal cousins share a character in their names, ok, but it genuinely isn’t necessary. Anyone with a single character name obviously doesn’t have one of these generation names, and by no means does every person with a two character name (especially female) have a generation name. If you’re doing an OC for an ancient Chinese setting (certainly anything before the year about 500), you shouldn’t use these generation names because it wasn’t a thing. Also, in a modern setting, even if such a generation name or name poem exists, it’s not like there is any legal requirement to use it (though there may be family pressure to do so).
As a further complication, some parents do the shared character thing among their children without it actually being a generation name per se because it isn’t shared by any cousins. Or, they have all their children (or all their children of the same gender) share a radical, which is a meaning component in a Chinese character.
If someone does have one of these shared character names, then their nickname will never come from that shared character; either they will be called by the full name or by some name riffing off of the character that is not shared. For example, I knew a pair of sisters called Yuru and Yufei with the same first character; the first sister went by her English name in daily life (even when speaking Chinese) while the second sister was called Feifei.
tl;dr Not all Chinese people have them, if you’re doing an ancient OC they definitely shouldn’t have them, and I honestly think the extra difficulty is too much for an author who (as the guide presumes) doesn’t read Chinese to begin with. For masochists only.