The president is referring to the doctrine of presumption of constitutionality. As Judge Sutton outlined in Thomas More Law Center, et al. v. Obama, et al:
The minimum coverage provision, like all congressional enactments, is entitled
to a presumption of constitutionality, and will be invalidated only upon a plain
showing that Congress has exceeded its constitutional bounds. United States v.
Morrison, 529 U.S. 598, 607 (2000). The presumption that the minimum coverage
provision is valid is not a mere polite gesture. It is a deference due to deliberate
judgment by constitutional majorities of the two Houses of Congress that an Act is
within their delegated power . . . . United States v. Five Gambling Devices, 346 U.S.
441, 449 (1953).
http://www.ca6.uscourts.gov/opinions.pdf/11a0168p-06.pdf
Obama, aware of the above precedent, is making an argument the ACA should be upheld accordingly, which is perfectly appropriate. Its the fundamental nature of Constitutional law and American jurisprudence in general: find the best evidence in support of your argument, then make the best argument possible.
Obama is not the Solicitor General. The SG makes arguments in front of the SC. And I dont recall the SG making any such argument. Congress' acts are only presumed constitutional. BUt that presumption may be overturned on a showing of the facts. Which is what has happened.