I would wager that all 50 have slight variations in their election laws, yes.
If Obama did NOT make an official legal name change along the way, all it could take is ONE Secretary of State to NOT certify election results to throw the whole electoral process off.
Local Secretaries of State don't certify presidential elections. The Senate does that based on the results of the Electoral College.
from Wiki: (http://en.wikipedia.org/wiki/Katherine_Harris)
Harris certified that the Republican candidate, then-Texas Governor George W. Bush, had defeated the Democratic candidate, then-Vice President Al Gore, in the popular vote of Florida and thus certified the Republican slate of electors. Despite the fact that the margin separating Bush from Gore was only a few hundred votes, with thousands of votes remaining to be counted, Harris ordered a halt to the count, freezing Bush's small margin in place.
... Florida state and federal law permit officials wide discretion in doing whatever it needed to permit voters to vote in such a way that their votes can easily be read and counted.