They're public accomodations like hotels. As per the Civil Rights Act:
The Civil Rights Act of 1964 prohibited places of “public accommodation” from discrimination based on customers' race, sex, color, religion, or national origin.
If forcing use of "Legal Names" is a policy that unfairly targets a group protected by that, look for a lawsuit eventually.
Every hotel I've ever checked into wanted to see my passport or credit card. So apparently a public accomodation does have the right to know your real name.
You most certainly can get a credit card under a pseudonym. If you're not doing it for fraudulent purposes, it's done all the time (very popular in Asian families). In some states you must file a DBA to do it, but you can.
It is a legally recognized right to use an assumed name in most jurisdictions in the US. In many of them, you don't even have to use it all the time, although some require that for it to be valid, and it becomes a name change then.
The Civil Rights Act of 1964 prohibited places of “public accommodation” from discrimination based on customers' race, sex, color, religion, or national origin.
If forcing use of "Legal Names" is a policy that unfairly targets a group protected by that, look for a lawsuit eventually.