According to this report:
Roughly one-third of Americans don't know that Obamacare and the Affordable Care Act are the same thing, according to a recent poll.
The poll, conducted by Morning Consult and published by the New York Times, found that 35% of Americans either believe that Obamacare and the ACA are different policies (17%) or didn't know if they were (18%).
This is not the first time Morning Consult has delved into the matter of "what's in a name?" In 2014, it polled on attitudes toward Medicaid expansion, with and without mentioning that is was part of the Affordable Care Act.
I suppose one can conclude that attitudes on an issue are fairly robust when support for a position doesn't vary much, depending on whether or not labels for the policy are varied. Conversely, if support does vary considerably as a function of labels, then people are likely responding more to the entity mentioned in the label (e.g., the president who supported the legislation) than to the specific policy itself.