They used her thumbprint to assess her ID. Since her thumbprint never changes over the years, they simply assume she's the Jennifer of the future.
6 seconds sound clip from the Back to the Future Part II movie soundboard.
You can hear this line at 00:25:21.779 in the Blu-ray version of the movie.
Quote context
[...]
- I am going to put this in the trash.
- McFly, Jennifer Jane Parker, 3793 Oakhurst Street, Hilldale, age 47.
- Forty-seven? That's a hell of a good face-lift.
- What the hell are they doing, Doc?
- What are we gonna say? That we're time travelers? They'd have us committed.
- She's clean. That means we take her home.
- Home? To Hilldale? It'll be dark by the time we get out there.
[...]