Assuming they hold it’s assur to own one, I can see their logic.
1. A passerby may think it’s theirs. It’s like walking into a treif restaurant (to use the bathroom).
2. Theoretically, they can commit the same aveiros on your phone as they could on their own device.
Whereas by you using it, none of these “maris ayin”, etc. issues exist. They otherwise have no problem with technology. So they’re more than happy for you to look things up for them as they, too, recognize the value and utility provided by your device.