what do people mean when they call a fictional character their 'son'?

Um. I'm not sure if I'm answering this correctly, but I'll try my best...

#1

People: Two fictional (Completely normal, usual style of writing)

Circumstance: Character of no relation to another, younger boy is calling him 'son.'

Reason 1: Its usually used this way to signify that they feel like a father to the child, or at least that they care for them in some way.

Example: I'm glad you're alright, son; I was worried there for a minute.

Reason 2: It's being used harshly. I'd think it's meant to be derogative, as in the one being renamed 'son' is being called a child who is not as wise as the person calling them that. Probably used in scolding another.

Example: Listen here, son. If you think you can walk in here, snap you're fingers, and all of life's problems will be solved, guess again.

#2 (May seem somewhat strange, but is still reasonable)

People: Writer and fictional character

Circumstance: The writer is referring to the character as his own son

Reason: I know from experience, that writers feel connected to the characters they create. Its a little strange, but can be used sensibly.

Example: (Asked to write a character's death) These are my sons! I can't just kill one of them!

#3

People: Reader and Fictional character

Circumstance: The reader enjoyed the story in which the character was present

Reason: (I'd hope) This would be at least somewhat jokingly. Otherwise, I suppose it could be simple, strong attachment to said character(s).

Example:

Person A: Come down for dinner.

Reader: I can't! I have to find out what happens to my sons!

I personally think that you would have no reason to call a character your 'son' unless you had a hand in creating or possibly editing the character. Otherwise it's kinda creepy, not gonna lie.