Legal adoption did not start until 1930.
Prior to that, a child or children would be taken in by close relatives (usually) or friends (occasionally).
If the girl was on her own and age 13, there is a strong possibility that she got financial help as a pauper. The surviving Poor Law records (which can sometimes be incredibly helpful) should be with the local archives; in the case of Montrose this would by the Angus Archives at Forfar (they have a website).
Gordon Johnson.