Mary-Kate and Ashley Olsen said that they were dizygotic (fraternal) twins. However they were born on June 13, 1986, and during that time, the only way that doctors determined if a pair of twins were identical or fraternal was by checking if they shared a amniotic sac or not. If they shared a amniotic sac, they considered them identical twins. And if they didn't share a sac, they considered them fraternal twins.
However, identical twins can have two different sacs. There's many twins who thought they were fraternal until they got a official DNA test done to them and found out they were actually identical.
Mary-Kate and Ashley Olsen might have not had a DNA test to actually confirm whether they are identical or fraternal. They just took their doctors word for it because they were in separate amniotic sacs. But identical twins (the egg) can split early on enough that they end up growing in their own amniotic sac, and would be assumed to be fraternal. Doctors make that mistake all the time, and almost every time when look-alike twins labelled as "fraternal" get a DNA test, low and behold they were actually identical the whole time.