I thought breast cancer also followed paternal lineage. My paternal grandmother and aunt, both died in 50's and my cousin is fighting a second occurrence (my age - 50's). I got myself checked, but breast center only asked about mom's side.

True. Most inherited breast cancers are associated to mutations in the genes brca 1 and 2. These mutations can indeed come from your father's side. People with suspicious family history like yours should visit a professional with expertise in inherited cancers. Good for you for being aware of this.
Maternal or paternal. Either of your parents can transfer cancer genes (i.e. Brca1/2), including breast cancer genes.
Both lines can pass. The breast cancer genes which have been identified so far are called "autosomal" meaning that they can be passed on through either the maternal or paternal lineages.
From both sides. Perhaps they keep asking you about your mother's history because you already have told them so much information about your father's family.
Breast Center correc. Statistically, it is only the maternal side of the family, and probably only first-degree relatives (mothers, sisters, aunties) whose history of breast cancer affects a woman's genetic risk.