This was a serious investigation, involving police in S. Dakota questioning someone about a crime committed in Alaska. Across state lines, so not a shoplifting case one might assume.The difference between being a friend, boyfriend, or coworker and being a spouse is that marriage grants someone 'marital confidence privilege', meaning they can refuse to testify about certain things. I assume you can see why then, her being married to what I assume is an alleged felon sort of might be considered a 'pertinent detail'.

Edit: Also, "Rapid City Police Chief Steve Allender says Newsome was uncooperative, and the marriage license was relevant because it established the residency and the relationship of the two women."

Both comments are obviously true, barring an emergency she should immediately have gone to assist the police, that's expected in the military and the police (many of whom are ex-military) certainly would know that and assume correctly that her delay was a sign of being uncooperative in of itself, though I'd bet she gave them other reasons to do so. While we reduce her to a victim of prejudice, let's keep in mind the police do not typically go out of their way to be nice to uncooperative spouses of criminals, nor should they just because she was gay.