I love this question.
When I read your sentence, I couldn't find anything wrong with it. It sounded natural to me. Yet when I looked up the word native in the dictionary, I couldn't find a definition that would support such usage.
I checked Collins, I checked Macmillan, I checked Wordnik, I even checked the OED. Over and over again, I found definitions referring to place and region, but definitions referring to time proved elusive. The more I searched, the more I became skeptical.
Then I tried searching for the phrase
"native to the era" in Google books, and on the web. Google initially reports 12,000,000 results found, but it doesn't take long to see that's a false number (the actual result is closer to 25 or 30). There are only three hits returned on Google books, and none of them show that phrase.
You wouldn't be the first to use the word to refer to time instead of place; one blogger wrote:
Generation Z is the first generation that can truly be considered native to the era of social networking and high speed Internet.
I would call this a trap word, one that sounds fine when you first hear it, but one that might prompt the careful reader (or the pedant) to say that you are uninformed. If you are writing, I'd avoid this usage and opt for a rephrase.
If you really wanted to stick with it, and were challenged, I did find this vague definition on the Wordnik page:
native (adj.) Naturally related; cognate; connected (with).
That might give you some rationale, but it doesn't seem to be a common sense of the word.