When “something” is used in a negative sentence, it often changes to “anything” as in the sentence below.

I don’t need anything. (Not something)
I don’t want any tea. (Not some tea)

Then why does this sentence work?

This is not something that I really want.

Is this because this “something” can be replaced with “the thing?”
Isn’t there a better explanation?

Apple

The rule of 'some' and 'any' does not hold when the pronoun is modified by a clause.

You are correct in your concept of 'something' here being like 'a thing.' What you are describing is something (a thing) that you really want. You are not saying that this is a zero; you are saying that it is not an object, not a thing, not some thing (2 words) that you really want.

You could also say, though awkwardly:

Something that I really want is not that.