It's called a dash. Dashes are asides. They're stronger than commas but weaker than parentheses. They can be on both sides of a phrase or start a phrase that ends with a period. The sentences they're in would still make sense if you deleted the phrase, but the phrase gives you more information.
For instance, in your first example, "my sense of self" is just a different way of saying "my reason for being." The addition of "my sense of self" just helps clarify what the writer is trying to say.
In your second example, there are actually two phrases that amplify the original thought. That's unusual, but there's nothing wrong with it. If you read it aloud, I think you'll find that it sounds natural.
Formal writing doesn't often use them, but informal writing, like your examples, does.
Does that answer your question?
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