I don't know if there were any official adoptions in those days but certainly children were fostered out if the mother was unable to care for any children.
Sometimes parents were named on registrations if an illegitimate child was born to their teenage child, however that does not seem likely with Henry's eldest daughter (from his first marriage) who married in 1875. The other daughters were still quite young.
Is it possible Alice was fostered out by Henry because Mary Jane nee PHILLIPS was unable to care for her children so the descendants of Alice believed whe was adopted???