Because it's saying Ich = du -- I am (the same as)(like) you. So nouns in the sentence get the nominative. That's true with all the forms of the verb "to be": ich bin, du bist, er,se,es ist, wir sind, sie sind, ihr seid, Sie sind.
So also "Er ist ein Mann," not "Er ist einen Mann."
He is saying that both pronouns Ich and Du are used as nominative case. Compare to "Ich mag dich" - I like you, the you here has been acted upon. But in the sentence "Ich bin wie du", the you here has not been acted upon, it was only used for comparison ie both in nominative form.
She says, when "sein"="to be" is used, you must use the definition after "sein" in nominative form. E.g "Ich bin (ein) Arzt" = I am a doctor. This example is not very good, because in this german sentence the article is not used. Another example: "Ich bin ein Mann, der ...", so "ein Mann" is in Nominativ form, not "einen Mann". Thus, "Ich bin wie du" ~ "Ich bin du" and nominative form is used.
It didn't accept "I am as well as you." it says that "I am similar to you" is the only correct solution. However when i hover over "wie" it gives "like," "as," and "as well as." I figured that "I am as well as you" would be the best answer because it sounds correct in English, I actually say it all the time to my friends...
I think the best translation of "I am as well as you." is "Ich bin genauso gut wie du." "well" is translated as "gut". The german sentence has the meaning of being as clever/intelligent/well in character as you. The German version with "gut" leads away from other possible meanings of "ich bin wie du." Meanings like "I am as fast happy as you." or "I have the same character like you" or "I have the same favorit hobbies like you", ... .