I agree with Sakura that motherhood is an intrinsic part of womanhood, if we see women as a group (which as feminists we have to do, otherwise whose rights are we talking about). Women are the only people who do motherhood. It is an issue particular to us - we are the group of people who give birth and breastfeed. That doesn't mean that women who are not mothers, are adoptive mothers, or who don't breastfeed are somehow less.
I don't want to be considered a person rather than a woman. I don't want to be seen as a person rather than a mother. What sex you are in not incidental if you have been pregnant and given birth. This is not some extreme situation where if we acknowledge pregnancy, birth and breastfeeding as important, the consequence of that is to let those things solely define us and demonise those women who haven't done it. It is not some either/or situation where women can only be seen as mothers or only be seen as people.
I can see why some people don't want to be seen as a mother - because people have negative ideas about motherhood and it can be detrimental to you at work if people are still seeing you as a mother in that situation.
But I am fortunate in that I work in a place where people don't have a negative attitude to me because I am a mother, so I can still talk about the fact that I have children and aspects of my children's life, what I have been doing with them and so on, and nobody will think less of me for mentioning them. One of my colleagues has a baby, and her baby is brought into work at lunchtimes for her to breastfeed in the lunch area, and there is no issue with that. I am a mother all the time, not just when I am with my children. And when I am with my children, I still have more interests in life than just my kids.
So with the childfree movement, what are the women in that movement asking for (of course there are men, but I am concerned about what the women want)? Are they asking for a society in which they are seem as being important, valuable people whose lives without children should be seen as worthwhile, whose lives, thoughts and experiences are of interest to others? Are they asking to be treated in an inclusive way, to have their choice not to have children to be seen as a valid one that they should not be called upon to ever justify or explain?
Of course they should have all of these things. It is ridiculous when they don't.
But there are some things that they can't have (and hopefully most in such a movement don't want). They can't ask for more public spaces not to have children in them (and consequently the primary carers - usually women) of those children. They can't put unrealistic demands on what behaviour children are going to exhibit in those public places (although clearly there has to be a balance). They can't expect primary carers of children - usually women, to not mention they have children, or to not talk about their children, or to show no interest in what those women are saying. Treating mothers in such a way is rather like how some people want gay people to behave - You can be gay, just don't really mention it, and certainly don't talk about the person you're in love with and spend a lot of your time with.
No person can live childfree. You can not have children of your own, but children are a part of society and a part of life. I know people who are not parents - they still interact with children and expect children to be part of life. So perhaps it is really the name that causes the issue.