This is a fair question, especially for anyone with light colored leather, suede, or fabric interiors who has had a bad experience with a spray- or oil-based product in the past.
The short answer: a properly designed hanging car perfume, built specifically not to leak, drip, or make direct contact with interior surfaces, should not stain anything, because the fragrance card itself never touches the seats, dash, or headliner when hung correctly. The entire point of the hanging format, as opposed to a spray, oil diffuser, or vent clip with exposed liquid, is that the fragrance stays contained within the card and diffuses into the air rather than onto surfaces.
Where staining risk actually comes from, historically, is with different product categories entirely: aerosol sprays applied directly to fabric or leather, liquid oil diffusers that can leak if tipped or overfilled, and cheap fresheners with unsealed fragrance oil that can weep or sweat in high heat. None of these describe a well-made hanging card, which is sealed and designed for exactly this use case.
Heat is the one variable worth being mindful of regardless of product type. Extremely high interior temperatures, the kind a car can reach parked in direct sun during summer, can affect how any fragrance product performs, including causing minor softening in some lower quality cards. A cosmetic-grade, well-constructed hanging perfume is built to hold up to normal automotive heat conditions without breaking down or leaking, which is part of what cosmetic-grade actually signals about the manufacturing standard.
If you do have unusually delicate materials, light colored suede or specialty leather, a reasonable precaution is simply making sure the card is not making direct, prolonged contact with a specific surface, easy to avoid with a slightly longer string length. Beyond that, a hanging car perfume is one of the lower-risk options available for exactly the concern you are asking about, especially compared to sprays or oils.