Correct or not, the doctor is confident that the attenuated virus in the vaccine is sufficiently attenuated that your niece isn't going to get the bad consequences of the wild type virus.
According to Wikipedia on the MMR vaccine, "10% of children develop fever, malaise and a rash 5–21 days after the first vaccination" With numbers that high, it sounds like statistically this isn't a serious problem.
Your niece is of course not a statistic, but at least based on what I know about medicine and biology, and what I've just looked up, the odds are strongly in her favor.
Skim comments to this Daily Kos diary for testimony from people older than me about just how bad things were before we had these vaccines, including of course the family and friends who didn't make it. My parents were born in the middle 1930s; they too attest to it.