Politicians always claim more credit for a good jobs market than they deserve and get more blame for a bad jobs market than they deserve. Politicians in general and presidents in particular have limited impact on the economy. Especially positive impact.
The jobs numbers have been improving, but the inflation numbers have been getting worse. This suggests that it has more to do with Federal Reserve action (or lack thereof). The money supply would seem to be growing faster than is necessary for real growth, causing some inflationary growth. To the extent that that is happening, current growth may turn out to be illusionary.
Prior to the election, some had asserted that a Donald Trump win would cause a stock market crash and a recession. That view lasted for about a day after the election and then the market started setting new records. But if that perspective had been true, we'd be headed towards recession already. And even that confuses things a bit.
Stock markets have limited impact on the overall economy. A rising stock market makes it cheaper to raise capital. A falling market means that raising capital is getting more expensive. But only a limited number of companies are raising capital. And the impact on other things like benefits is limited since the stock market as a whole is moving.
Since the election, some people are reacting as if things that they expect Donald Trump to change are going to happen. Or that Donald Trump is going to block changes that they had expected Hillary Clinton to make. That could mean that some of the growth could be credited to Trump.
The economy went into recession around 2007, after the Democrats won the Congressional elections. It was recovering around 2009 and Republicans won in 2010 for 2011. So should we credit the 2011-2015 period to Democrats (presidency and Senate majority)? Or Republicans (House and Senate filibusters)? Does it matter that Republicans also took over a majority of the governorships and state legislatures in 2010 as well?
This also changed again at the beginning of 2015, when Republicans took the Senate. With the two year lag hypothesis, that would start being felt in 2017. And a two year lag is just a convenience. For example, the housing crash had its roots in the late 1990s. George W. Bush didn't stop or reverse those trends in the 2000s, but he didn't start them either.
We'll never have a real answer though, as there's no way to falsify a hypothesis. We can't rerun the 2007-2009 period with a Republican Congress. We can't rerun the 2009-2011 period without Democrats in control of all three of the presidency, the House, and the Senate (including months with a Senate supermajority). We can't rerun the 2011-2015 period with Democrats retaining the House. So we can't say what would have happened in those cases.
It should be obvious that to the extent that actual policies matter, they can't matter if they haven't been enacted yet. Only the perception and anticipation can matter. So if Trump deserves credit now, it's purely for perception and anticipation. That said, it's entirely possible that the perception and anticipation matter more than the actual policy. Perhaps business people were simply waiting for a president who empathized with their problems to feel secure enough to hire. But that would be nearly impossible to verify or disprove, particularly on the margins.