I suppose you're right that he could be considered part of the press, and there are no laws prohibiting the sharing of government secrets. But I'm hesitant to proclaim anyone with a blog or government secret as a true member of the media.
There actually are laws prohibiting the sharing of government secrets. Lots of them. Bradley Manning, the suspected informant, is likely to go to jail for this incident.
But once published, they are published.
As for whether random bloggers are members of the media, I agree that it isn't a simple cut and dried situation. However at this point wikileaks is so prominent, and is so frequently part of the information infrastructure that more traditional media provides, that I believe it clearly is media.
Well, there's a difference between sharing and publishing though right? In this case, Bradley "leaked" them = illegal. Assange "published" them = freedom of press?
In what way, at this point, is Assange not part of the press?
Remember, a large part of the Constitution is meant to provide checks and balances to prevent government from being able to abuse its power. The proper Constitutional role of the press SHOULD be to inform the public on the actions of the government so that the public can get properly outraged about things they do not agree with.