In cases like this, someone or someones, generally a married couple would raise a child as a member of their household, sometimes as their own child, sometimes also as part of their legal familia. The problem is, it seems to have been uncommon enough that the term alumnus/a, here and elsewhere translated as “foster child,” covered a rather broad range of situations. Essentially it covered any child raised by people not their birth parents and who didn’t go so far as to legally adopt them.
To some extent, these relationships also weren’t incompatible with the possible slave status of the child. There was a term for a child born into slavery within the household: uernae. Such children might also be considered alumni of the household though, especially if they were later freed.
One example of this might be the case of Petronia Iusta. Her mother, Petronia Vitalis, a freedwoman of Petronius Stephanus and Calatoria Themis, had left Iusta to be raised her former masters for several years. Eventually she returned to claim her daughter and pay them for the cost of maintaining her. The subsequent legal dispute over Iusta’s status as freeborn or freed is interesting, but not relevant here.* Iusta, whether she was born slave or free, was raised as a child of the household and was therefore considered an alumna, a foster daughter. Because of that, while she wasn’t necessarily entitled to any inheritance, not having been adopted, she did have a certain social connection to them beyond being either their freedwoman or the daughter of one.
*Basically, Vitalis claimed that she had been free at the time of Iusta’s birth and that Iusta was therefore freeborn. Calatoria Themis, on the other hand, claimed that Iusta had been born a slave and was later manumitted by Themis herself, making her Themis’s freedwomen. Neither side had the documentation to back it up.
Rawson, Beryl. "Adult-Child Relationships in Roman Society." In Marriage, Divorce and Children in Ancient Rome, edited by Beryl Rawson, 7-30. Oxford: Oxford University Press, 1991.
Weaver, P. R. C. "Children of Freedmen (and Freedwomen)." In Marriage, Divorce and Children in Ancient Rome, edited by Beryl Rawson, 166-190. Oxford: Oxford University Press, 1991.