Bush's made an empty campaign promise then Obama kicked the idea so far down the road it was totally out of sight.
But five years? That's doable insane. Like, get Elon Musk, Jeff Bezos and NASA in a room ask them who wants this more.
1) Bush's does not seem to have been a campaign promise. While the vision was proposed in an election year, it also received Congressional funding - $16.2 billion - and legislation in the same year.
Note that Trump started his reelection campaign when he entered office, so anything he says about the future can be regarded as a campaign promise, yes?
2) There does not appear to be funding behind the Trump administration's proposal. Quoting the NYT piece, "an accelerated pace has not been evident in the Trump administration’s NASA budget requests to Congress, raising many questions about how it will be possible for the agency to accomplish this ambitious goal." Much less the White House pressure that the Bush administration used to get their full budget through.
(Quoting https://web.archive.org/web/20050913093403/http://www.space...., "The White House, which had threatened to veto a bill moving through the House during the summer that would have cut $1.1 billion from NASA’s request, leaned on appropriators during the closing days of the budget battle to make NASA a high priority.")
3) I see no indication that Trump prioritizes this policy anywhere near as high as Bush did.
As to "doable insane" for going to the Moon now, note that Bush announced in his 2004 speech "Our second goal is to develop and test a new spacecraft, the Crew Exploration Vehicle, by 2008, and to conduct the first manned mission no later than 2014."
That was 4 years for a new spacecraft. Certainly that's more doable and sane than going to the Moon, which was Bush's third goal. Yet it wasn't done.
No matter which of Musk, Bezos, or NASA might want it more, how much money do you think they'll need, and where is that money?