So, we're gonna discuss 名 míng 字 żi 號 hào, as pertains to my understanding of it according to my area of romancelandia/wuxia/Taiwan conventional SFFR usage, okay?

All the qualifiers because apparently certain c-dramas take certain liberties etc.
China is not a monolith etc.

First off, you have the 名, or what is commonly referred to as the birth name.

This is what you are named when you are born or shortly thereafter.

This is where generational names come into play.

You do not have separate generational names for boys /girls.

In a posher family, sometimes they have a poem they like to use. This is so you can tell how people are related just by looking at their name.
Often also you can easily go "oh, that's the Chen family from Lanling because that's the generational name of that branch".

If the poem goes, 雲翔亦龍，鳳舞風終，and you have three people named:

Lin 雲 xiang
Lin 亦 hui
Lin 龍 yao

Then you know Lin 雲 xiang is three generations above Lin 龍 yao and you make your respects accordingly.

In case it wasn't clear - you absolutely do not share names with your parents, grandparents, etc etc.

Major taboo.

In fact, the word for taboo is literally "name of your ancestors“，so the whole namesake concept doesn't apply here.

Nicknames or 小名 （literally little name or diminutives).

This is where it gets tricky because we no longer live in ancient times and I'm not a scholar, but my-culture-by-osmosis brain informs me that we do not use nicknames on 字。

The birth name gets nicknames.

To clarify, the birthname gets nicknames like ah-name. Or name-name 軒軒. Or xiao-name. 小軒。

Punny nicknames are another situation entirely

So for everyone trying to cute-sify 字, that's, as far as I can tell, Not A Thing.

Because the whole point there is aspiration+dignity.

Whether the 名 has one or two words is entirely up to the region.

It's uncommon in Taiwan for people to have single character names, frex, but it used to be very common over in China.

Also depending on the region, it can be seriously grandiose and aspirational.

Taiwan, due to History reasons, for a period had a lot of boys named 復國 or literally "get back the homeland".

建國 or "establish a nation" was also really common.

Noting here that the 名 has the aspiration and hopes of the "parents".

乳名 or literally milk-name is slightly different from nicknames.

Sometimes this is where the "call your baby trash and hope the gods ignore him" thing comes into play.

Or "meatball", "dumpling"..."yogurt"...

Man, people are WEIRD. I know people like to chew on babies but...

Your parent/grandparents/aunties/uncles, etc call you by your milk-name.

Your siblings or cousins would call you by a nickname.

In case it wasn't super clear yet - who calls who what is highly structured and has to do with respect/allowed intimacies.

Flogging the nickname thread again because why not. https://twitter.com/KatjeXia/status/1086471039754821632?s=20

Finally, moving onto 字。

Honestly not a scholar, so all this is cultural osmosis, complicated by netizens and their adaptation...

BUT.

字 is chosen by the person themself, usually after they are considered an "adult" or "old enough to know better".

The 字 is the aspiration/hopes of the person themself, and it does often have links to their birth name.

It can even explain or expand upon the birth name. It's rare that there's no link entirely, and if so, then that in of itself is a story.

So, say my name is 映澄 or "reflects clarity".

My chosen 字 might be 漣斂 or "reined/restrained ripples".

And you see here that there's a link - a lake without ripples better reflects things clearly.

So the implied goal is "discipline in search of truth".

Because my 字 reflects my hopes/dignity/aspirations, it would be inappropriate for people to add diminutives to it along the 小/阿/repetition family.

Single-character-ing it is debatable according to how well we know each other.

A loved one calling me 斂 might be allowed.

Very "might", because the meaning of the entire name can be summed up with that one word, or it could function as a reminder when used that way.

But it would be uncommon and not something a rando could/would do.

Your reminder here it is perfectly fine to call your friends by their one-character names or to shorten their names to one character - but it's very much a BY CONSENT and possibly a "do you want to hint that you want to bang" sort of case by case situation.
YMMV.

One thing to keep in mind is that the 字 is, to a certain extent, where the person has a little more flexibility to be themselves, or to explore themselves.

They are unlikely, unless their identity is bound up with their brothers, to choose similar ones to their siblings, frex.

Which leads us to the 號, or as we know it, "the ancient equivalent of twitter handles where people call themselves stuff like THAT BITCH FROM THE NORTH".

Here is where your creativity shines. Or does not.
The 號 can be grandiose, like The EAST NEVER FAILS.
Or humble.

Unlike Twitter handles, people mostly didn't change their 號 that much, but some people still had two or three.

Essentially, the more real-world famous you are, the more of these you can carry off.