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What name or word would be given for the examples below, metaphor?

1. A mother calls her child cat, or tiger


2. The wife calls her husband "Hey Baby".


3. You're my moon.


4. A very muscular person might be called "He's solid".

Basically there's extreme likeness to the other object and the person gets called by that name.

But this one turned into a weird argument; I have a black friend and he loves to be in the dark. He likes having lights turned off, so I said to him "... Yeah dark guy." But this is only following the above examples as well.

When a person is given a name corresponding to an attribute of that person, or a thing is named by something closely associated with it, it's called "metonymy." Only the last of your examples seems to be metonymy, however, and even then it's not an obvious example of it. So I'm not sure if that's what you're looking for.

The first three seem to be examples of a "pet name" if it's regularly used, or a "term of endearment" if it's used only occasionally. As I understand and use the terms, a pet name is a term of endearment that becomes a sort of private nickname.

(The third is a clear example of metaphor, but I'm guessing you weren't asking about that.)

Your first three examples are of terms of endearment.

The fourth is simply a description, rather than a name, although "Solid" alone could be used as a nickname, based on physical characteristics.

What you called your friend is possibly in the category of being a nickname, based on habit or personality. Unfortunately, when spoken, your chosen term of "dark guy" sounds very like "darky", an ethnic slur that was historically used against black people (and probably still is). I can understand why he was unhappy to be called that.

Some general advice: do not give people nicknames. They're rarely appreciated by the recipient. If someone has a nickname that they're happy with, they'll tell you it.

A more intimate nick-name like this is a pet name

an informal name given to someone by their family or friends

https://dictionary.cambridge.org/dictionary/english/pet-name

I would go with pet-name as the above answers suggest, but if you wanted a single word, you can also use Hypocoristic, which can be both an adjective describing the name ("she liked the hypocristic name he gave her") or a noun for the name itself ("he would always call her by her hypocristic"):

Hypocoristic

Of the nature of a pet-name; pertaining to the habit of using endearing or euphemistic terms.