Unlikely. Fortunately, the card doesn't have an address or any other number that could be of wider use to a potential fraudster, such as an SSN or driver's license number. This is why many people like showing a Nexus or GE card at TSA since the card contains the minimum amount of information necessary to be REAL ID compliant. Furthermore, upon replacement with a new card, the old one will be deactivated.

Is the junk mail you receive in your name, or is it addressed to the current resident or a direct mail advertisement (without your address)? If things aren't arriving that are addressed to you, I'd make sure that delivery is setup for your address. Did you double check your mailing address on file on the dashboard for your trusted traveler membership?