No, Trump selected him to be the chair, he didn't merely renominate him.
But, in any case, Obama nominated him but did not choose him so much as he choose not to violate long-standing precedent as to how the seats that cannot (because the 3 members of one party limit has been reached) be from the President’s party are selected.
Now, he could have violated that precedent, but that wouldn't have likely gotten someone else into the seat, just given up any chance of Senate confirmation.
Obama still made the decision to nominate him. Plain and simple. And Trump did renominate him for another 5 years. Yes, he elevated him to chairman, but he still gets the same 1 vote, so it doesn't matter. Not quite sure what point you were trying to make.
But, in any case, Obama nominated him but did not choose him so much as he choose not to violate long-standing precedent as to how the seats that cannot (because the 3 members of one party limit has been reached) be from the President’s party are selected.
Now, he could have violated that precedent, but that wouldn't have likely gotten someone else into the seat, just given up any chance of Senate confirmation.