It's ok to have ground rules. Such as, "I know you are aware I see your cousin Sally. Please know that anything you say about Sally will be held confidential and similarly, all of Sally's treatment information will be confidential as well."

However, in general, a client can say whatever the fuck they want and that doesn't impact your ethics at all until it becomes an issue of treatment being compromised and /or reportable offenses. Provider ethics is the professional responsibility toward the client and the profession.

I think I get what you're saying. You don't want your established therapeutic relationship with client A to be tainted by new information from client B. However, once you've accepted client B into treatment, you have ethical obligations to them as well.

Also, depending on your location, it's also not rare for those who know one another or share (even intimate) relationships to share clinical treatment providers at all levels. Many areas of this nation (US) remain under serviced despite a saturation of professionals in the field. It isn't ideal to treat two related individuals but it can be done in certain situations. Referral is optimal but not always an option.

I believe in the circumstances where you wouldn't want to continue treatment because of this issue, it's likely better headed off before treatment starts. In pre-session or intake paperwork, "please know if you have any immediate family who use me as their primary psychological provider, I prefer to refer you to my exceptional colleague XXX who would be just as skilled in your treatment."

You seem to be focusing a lot on the TL;DR but I hope this message reaches you and gives you some ideas on how to approach this issue as you transition from student to practice.