Isn't it a common practice to find nicknames for pieces of legislation? In that case, the Republican found one that stuck. The name "Obamacare" is nothing more than a portmanteau of "Obama" and "Health care", and as such it does not entail any political message per se, except that it's supposed to name an health care legislation in which Barack Obama was particularly involved.
So, calling the use of this word "propaganda" seems highly incorrect.
In fact, IIRC Barack Obama himself appropriated the name and said something like "I don't mind this to be called Obamacare, because I do care" (I'm paraphrasing).
It seems to be increasingly common to make the name of the legislation propaganda in itself. Naming the ACA 'affordable' would be an immediate example - a less contentious example would be "USA PATRIOT" (Uniting and Strengthening America by Providing Appropriate Tools Required to Intercept and Obstruct Terrorism Act), as attacking PATRIOT immediately sounds politically worse than attacking state surveillance.
Countering 'Affordable' with 'Obamacare' was a very successful propaganda win. Affordable sounds good, Obama sounds bad (to the intended audiences). That you can realistically ask people to take sides between the two (whether they prefer Obamacare or ACA) illustrates this perfectly (if Kimmel is a source, https://www.youtube.com/watch?v=N6m7pWEMPlA)
For a large portion of the population, "Obama" is/was an inherently negative word.
While it's true that the word simply indicates it was "his" plan, this is also intrinsically linked to extreme negativity and calling it Obamacare was extremely loaded.
I can understand why this could be seen as propaganda.
In response to your edit, I think this was a good way for Obama to take things in stride. But it doesn't change the inherent negativity for most.
So, calling the use of this word "propaganda" seems highly incorrect.
In fact, IIRC Barack Obama himself appropriated the name and said something like "I don't mind this to be called Obamacare, because I do care" (I'm paraphrasing).