I was a step-mom but the birth mother had died. My step-son was 6 when I married his father. He never did want me to hug him but he liked praise.
I adopted my daughter who came to me as a foster child when she was 7. I was with her and her birth mother at least twice/month. I put my arm around her shoulder whenever she was near me. I complimented her and her birth mother whenever it was appropriate. Her birth mother was angry with me at first but gradually she warmed up to me and our meeting went smoothly. Of course it was obvious that she had something to gain by being nice to me. It's not so obvious to some mothers or step-mothers who let their feelings get in the way of what is best for the child. I agree that it is best for the child to have many people love her and show their love. To hold back because the other mother may not like it is begging the question.
I'm glad that you've asked this question and are concerned about what is best for your step-daughter. Talking to the mother is a good idea. You have nothing to lose even if she responds in a cold manner. I also suggest that you talk with your step-daughter. At 6 she is aware of a lot and is probably wondering herself what she should do. She may feel that to hug you would upset her mother. But that isn't necessarily so. Her mother may not mind. She just isn't sure and doesn't want to take a chance. She may also be afraid that you won't want to hug her in front of her birth mother. She is caught in the middle.
My suggestion is to let the mother be the first to hug and congratulate and then to reach out your arms while also praising her and see what happens. She may be just as anxious about your reaction as she is about her mother's and would welcome you making the first move. If she doesn't move into your arms, pat her on the back or on the arm and let her know by your words of praise and your closeness that you love her.
I feel sad when I hear step- mothers say that they don't hug their step children in front of the birth mother. It seems that they're acting on an assumption that may not be true or which could possibly be overcome. Everyone in that situation feels uncomfortable and doesn't know what to do.
My philosophy is to do what seems natural and see what happens. If this were a child of a friend you would hug her. The child is likely to just be standing there waiting to see what happens.
The birth mother's first reaction may be negative but if she doesn't tell you to not do it or her daughter to not let you do it then keep on giving hugs. This may be what needs to happen to break some of that tension. When we hesitate and do not hug we're giving the message that we don't think we have the right to give that hug and we do. As nearly everyone agrees, even many jealous birth mothers, every child benefits from being loved by lots of people.