There is no one single way in which people in monogamous relationships live. There are many different ways that such relationships can work. The only thing that they all have in common is that the two partners are sexually faithful (and, presumably, emotionally faithful) to one another. If they truly are monogamous, they will remain faithful in these ways.
Outside of this, however, there are different kinds of monogamous relationships. For example, there are monogamous relationships in which the partners have children and there are those which do not. Partners with children live very different lives than those without.
As another example, there are relationships in which the two partners are equal and relationships in which they are not. In some heterosexual relationships, the man and the woman see one another as equal. They see themselves as partners. However, there are other heterosexual relationships in which there are differentials in power. One partner may make more money than the other. Society may stipulate that the man is the leader in the relationship and the woman is the follower. Such relationships will be very different indeed than those in which both partners feel that they are equal to one another.
Thus, it is possible to have many types of monogamous relationships. The only thing they really have in common is that all of them include sexual fidelity between two partners.