I think that saying is more descriptive than prescriptive–it’s not what ought to be, but what often is.
Generalizing wildly, I think the reason is that a lot of the time, women have stronger bonds between each other and work harder at their relationships. So in a mother-daughter relationship, both women are often making an effort to work at the relationship, while with a mother-son relationship, often only one party is making an effort. With a father-son or brother-brother relationship, the worst case scenario is that neither one is doing the work. (For a less charged example than MIL-DIL, my sister and I frequently call each other. My brother and I don’t. In fact, most contact is between me and his wife when we discuss travel and gift arrangements via email. If there were to be more interaction between our families, I would have to be the one to do it.)
A lot of the time, the wife is in charge of managing and preserving the family’s relationships. For instance, I often ask my husband, “Have you talked to your parents lately?” whereas my husband never asks me if I’ve talked to my parents lately. So, it’s not necessarily a question of the new DIL doing something evil if the mother-son relationship dwindles a bit–it may be more a question of her not doing something (not encouraging her husband to talk to his family).