I'm sorry I'm not who you thought I was.

It looks like a relative pronoun to me.

I — S
'm not — V + neg
who you thought (that) I was — relative clause modifier

I'm sorry I'm not who you thought I was.
If who is a relative pronoun, where is its antecedent?
Compare: I'm sorry I'm not the person who you thought I was. (Here, the antecedent is person.)

That's a conundrum. "Who" has two different antecedents, but they both happen to be "I". "Who" is the predicate nominative in "I'm not who", and it is also the predicate nominative for the second clause, "I was who". It is also the relative pronoun for the second clause.

That sort of multiple role is usually wrong, but this sentence is natural English. English is not strictly bound by the

For simplicity, I'll take out "I'm sorry" and "you thought".
Now it's: I'm not who I was.
Now compare: I know who I was.
Is the latter who also a relative pronoun or a question pronoun?

It's still a relative pronoun. It's an interrogative pronoun in this question: "Who was I?"

He asked, "Who is he?" -- Who is a question pronoun.
He asked who he was. -- who is a relative pronoun??

Yes!

Could you please provide me with a non-interrogative sentence that includes "who" and uses it as a question pronoun?

No, I can't because "interrogative" just means "of, pertaining to, or conveying a question." A question pronoun is an interrogative pronoun.

What about this sentence then?
I wonder who that letter was from.
See the link: http://oald8.oxfordlearnersdictionaries.com/dictionary/who
And some other links: