You do have a legal obligation for anything financial where KYC laws apply, and I wouldn’t try that with airline tickets, either. You might also have issues around breach of contract if you’re lying about something which might have affected their willingness to allow service – this is likely not to go beyond cancelling your account but you should think about the magnitude. Nobody’s going to care about your grocery store loyalty program, but if there’s money or copyrighted material involved you might want to weigh your willingness to be a legal test case.
An alternate or temporary card number isn’t providing false information. It’s also less sensitive since they know not to store it and, at least for Americans, your liability is lower - you can rotate numbers easily, but not your PII.
