Is it obligatory to distribute aid that exceeds one's needs, or may one keep it, and is taking more than one's needs considered transgressing the limits set by Allah?

The matter of aid is to be considered by the authority responsible for its distribution. If the responsible employee states that the four cartons are for the family, or permits taking them without infringing upon anyone's right, then there is no harm in the family taking all of them. If you share with needy neighbors, you will be rewarded, for the Prophet, peace and blessings be upon him, said: "He is not a believer who eats his fill while his neighbor beside him is hungry," and he also said: "He has not believed in me who eats his fill while his neighbor beside him is hungry, and he knows about it."