Is the father entitled to claim a portion of the inheritance of his maternal aunt who raised him from childhood, or is he not entitled to that?

Custody and upbringing are not considered among the causes of inheritance. The causes of inheritance are limited to three: marriage (Nikah), wala' (clientage), and kinship (Nasab). The nephew (sister's son) and the niece (brother's daughter) do not have a share in the inheritance of the maternal aunt or the paternal aunt because they are not legal heirs.