Anyone can choose to use a card instead of a Magic Band. That choice is actually part of official privacy policy for MyMagic+. When you use the Magic Band, your information gets used to customize some attractions (like It's a Small World) and to trigger photos on other attractions.
Since some people are not comfortable with their name being displayed, they can avoid these enhanced experiences by using the plastic card. Since the card doesn't have a battery like the Magic Band, it can't be read by the long-range readers. That protects the guest's privacy.
For more specific information, here is the My Disney Experience - Frequently Asked Questions page on the Disney website:
https://disneyworld.disney.go.com/faq/my-disney-experience/my-magic-plus-privacy/