As I understand it, "mommy-and-me stuff" refers to something the speaker usually does with his/her mother and which has a special meaning for them or an appointed time to do it, as opposed to anything else.
In AmE, "Mommy and me" refers to any type of activity or "play group" that involves both mother and child (usually the child is an infant or a toddler). For example, if the local YMCA pool states that every Monday from 8AM to 9AM is "Mommy and Me Swim", that means that the pool will be populated with mothers and children. A "Mommy and Me" exercise class is one that involves mothers participating in aerobic exercises like "power walking" while pushing a stroller. There are also specialty "mommy and me" retail shops that sell both maternity clothes and infant wear, but it is not a trademark or chain.
Dear Rover,
I thought the context might not help. Indeed I assumed it was a term commonly used in AmE. However, this is the limited context:
I know men that stay home and work in the home to support wives with careers And it's hard. When I go to the Mommy-and-Me stuff and I see the father there, I notice that the other mommies don't play with him.