Welcome to Rootschat.
You are right, there was no formal adoption before 1927, so there are unlikely to be any records. It is a bit unusual, as far as I know, for a child to be given their father's surname if he is not mentioned on the birth certificate and the parent's aren't married. Not illegal, as I believe you can give a child any name you like, just unusual!
Something a little odd re the entry on Freebmd, the initials are in brackets i.e [F P] none of the others on that page are. I wonder if that is significant?