Both options are correct.
That's the classical explanation to your question would be that it's short for "Jag vill vara som du är", the verb is understood. Just like in German where they would use "du" and not "dich".
Although nowadays most Swedes would treat it is as a direct object (like in English) and say "Jag vill vara som dig". I've heard (don't remember where) that 90% would say "dig" in this case today. Don't know whether it's true, but it's a vast majority for sure.
English can have "understood" words excluded from the sentence also. People will argue with you if you say it's correct to say things like "You do it as well as I [do]" while others will argue when you claim it's correct to say "You do it as well as me".
The debate is centuries-old. Both ways are acceptable, like in Swedish, so don't let anyone tell you otherwise :)
Yes, but different things are understood in different languages. In English you wouldn't say "*you want to be like I", it has to be "you want to be like me" or possibly "you can't to be like I am". In Swedish you have the choice "du vill vara som jag/mig".
Btw it's pretty funny in French when they say "t'inquiète!" (meaning "worry!"). In this case the negation is understood and therefore it means the complete opposite. which I find very odd. In English (and all the other languages I know) you can't skip the negation if you want to say "don't worry!". If you just say "worry!", it sounds quite worrisome.
And there's a notable idiomatic difference in that English can choose to express wanting to change a current state where Swedish prefers future deviation.
In other words: phrases like "Do you want to be my girlfriend?" can translate well into Vill du bli min flickvän?
Yes, "Vill du bli min flickvän?" is correct.
It probably goes without saying, but if the girl already is "my girlfriend" you have to say "Vill du vara min flickvän?", because there's no transitional phase as the verb "bli" requires. Like if you have a scenario in which you're about to break up "Do you want to be my girlfriend or break up?".
The question makes me think about this song https://www.youtube.com/watch?v=VpQhiqBFpao. Not really funny, but helps you remember :).