I agree that common sense would dictate that the team's governor should sign off on it before it goes to the league. However, let me give you a case that's kind of similar:
When a college professor comes up for tenure, there's a committee that meets to make the decision, working with people in the candidate's department. (The exact system depends on the school, so I'll just say how it works where I teach.) Eventually those people -- the ones who are best qualified to make an informed decision -- decide what to do. If they want to keep the candidate, the committee passes that decision to the college president, and it's up to him to approve it.
At that point, the decision is announced. But technically it's not final yet, because the trustees have to sign off on it. Generally, their acceptance of the committee's decision, plus the president's approval, is routine, but you never know. Something strange could come up. Most people at the college don't even know who the trustees are, and it would make no sense for them to overturn a decision -- but since they oversee the college's money, the final call is theirs.
Since we don't know how the NBA works, it seems possible to me that it's a similar situation with respect to trades. Remember, most owners are pretty hands-off, and sometimes the "owner" is a faceless corporation, like the Knicks and MSG. I can imagine that, since it is the GM (or president of basketball operations, or whoever) that is responsible for engineering trades, it's that person's signature that closes the deal before it's sent to the league for approval. It could be that, at that point, the league then contacts each team's governor -- whoever it is -- just to check, "You're good with this, right?" And since the governor almost always says, "Sure, whatever," no one really thinks much of this step. It's like the college example: the GM is the committee who makes the decision, the league office is the president who approves it, and the governor is the trustees who ultimately sign the checks.
Of course I don't know if this is really how it works, but it strikes me as possible. It wouldn't be a very logical system, but who knows.
