Nah, most likely not. If you feel comfortable around your lawyer, then you should have no problem discussing their relationships to your label, your publisher, etc to avoid a conflict of interest.
think of a lawyer the same way you do a therapist. in your eyes, the most important thing should be is how they can help you.
if you somehow are friendly with your lawyer, developed a relationship, etc and you're not 100% positive that they're in your corner, and your corner only, then you're getting played.
A lawyer works for you, not the other way around. fukk comfort or a relationship. You get paid off my success. Work for my best interest and yours.
And no, it's not the most normal thing in the world. Most musicians are smart enough to at least have a lawyer that is independent of any label. I'll give a lot of things flak, but fukking with a lawyer that is on call with your label is stupid shyt 101