I don't want to put words into the OP's mouth with regards to his assertion about Schmidt, but given the loose wording: "basically shadow president", it's fair to say it isn't meant literally, and it usually comes with a negative connotation and to imply that Schmidt was so deeply involved, from a standpoint of strong biases in favor of Google and the obvious potential for corruption in participating as an advisor to someone (Obama) who doesn't have the same grasp on technology — and the extent and length of Schmidt's involvement throughout Obama's terms.

It could be said that Schmidt influenced important decisions in the tech realm, to a degree nearly equal to executive authority, because it presumably (and greatly) outweighs the opinions of the other heads of Big Tech, so long as Obama was naive enough to agree with him on key issues he didn't fully understand.

Of course, anyone could assert this about an advisor to a President depending on the President's level of knowledge and outright willingness to apply their advice, even if in spite of fairness, rights, laws, or precedents.