Many people say that Serana was already asleep during the Second Era due to a phrase from her involving the Dwemer. However, she was born and lived during a period without an Empire, which may imply that she was born before the foundation of the First Empire or during the ESO period.
None of that is to say that the Devs/Writers can't backtrack or contradict what they said earlier, but its not quite as cut and dried as the first quote makes it sound.
When you add in the Skyrim: Dawnguard dialogue from Durnehviir and the timing of Vyrthur's prophecy that kicks the whole thing off, it gets really hard to argue for a 2nd era date for Serana going to sleep. Certainly the Skyrim writers did not have ESO: Elsweyr in mind when they had Durnehviir date his entrance to the Soul Cairn and the bargain to guard Valerica as a time when "The dovah roamed the skies, vying for their small slices of territory that resulted in immense and ultimately fatal battles."