So why is it that the Argonians refer to other newly-born Argonians as "hatchlings?"

Before you answer that question, STOP RIGHT WHERE YOU ARE. I'm aware that Argonian children hatch from eggs, so I know that the term "hatchling" is a valid way to describe their young. But when you consider that Cyrodiilic is not an Argonian's first language, you have to wonder why they'd take to using that word to describe their young, rather than using a more familiar word such as "baby," "infant," or "child." You have to wonder still more when you realize that they use the word to describe all kinds of children, including human children, which obviously do not hatch from eggs.

So, now you may begin pontificating the Argonian etymology of the word "hatchling."

How do you know humans to not hatch from eggs in TES universe, I say just about anything is possible... just kidding
On a more serious note I would say it is probably just a habit of theirs, kind of like how Brynolf says lad to guys all the time.