There is no surname shown on a birth regsiter entry for the child until 1969 (in E&W). Entries are indexed in one or both of the parent's surnames depending on their marital status. So this is indexed twice, not registered twice.
For an unmarried father to be named on the entry (after 1874) he would have to be present to sign as a joint informant with the mother .... sounds like he did that. That is still the law today.
It is possible for an unmarried father's name to be added later by applying for a re-registration, but that creates a completely new entry, it doesn't change anything on the original one.