Do you know exactly who your brand is? vs. Do you know exactly who is your brand?
If I consider that your question is only about the correctness of one of the questions above, I would have chosen the first one and replaced the "who" with a "what".
Do you know exactly what your brand is?
The question is correct now from the grammatical point of view. The reason I chose the first question lies within explaining a vast 'sea' of grammar, but this much is enough to know that if the question that is asked about something, has other question-like phrases in it, none of those phrases are going to be in the structure of the question.
For example, take a look at your own question. What is the main "idea" you've asked? The main part of your inquisition is "do you know?", not "what is brand?". Just think about it. Your friend is going to answer "do you know?" first and then say something about the brand.
So, according to the bold statement, "do you know?" should be in grammatical question structure and "what is brand?"shouldn't be. So we have "what your brand is", just like a normal sentence.
And I chose "what", because "brand" isn't a someone. A relative pronoun for 'things' is what, and it's the appropriate pronoun here.
On the other hand, and especially when I read Damkerng T's comment, I realized you might have asked another question, meaningfully.
Maybe you asked your friend about something you both have, for example T-shirts. In this case, if I were you, I would've asked something like:
What brand is yours?
Do you know what brand yours is?
Notice that if this was your question, then the use of "exactly" is not really sensible. There are other possible means of this question, but this one was what I supposed, since it's the most plausible and common possibility. Hope I've helped.