In such situations, what can happen is the mother has custody of the child during the week, while the father gets to spend one or both days of the weekend with him or her.
Phrase is especially true if the mother is remarried or has a boyfriend, hence the child's father figure during the week. This makes the biological father the weekend dad.
Father- "Stop calling me that! We're going to the water park."
Kid- "Thanks, weekend dad!"