It was very common for children to assume the surname of their stepfather when their widowed mother remarried.
When they themselves married, they might do so under either surname. "Father's Name" on the certificate could refer to either. Birth surname seems more common.
I've also come across a family who used the wife's maiden name at all the censuses. Most of the children used the husband's real surname at marriage, though one of those then carried on using the surname he had been brought up with.