Is it permissible for someone who has slaughtered two aqiqahs or two udhiyahs to eat all of one and donate all of the other, or is it obligatory to donate a portion from both?

It is obligatory to give a portion, even if small, of the sacrifice (udhiyah) and the offering (hady) as charity, based on the apparent meaning of the texts and hadiths. Eating from it is commendable. If one does not give any of it as charity, they are required to buy some meat and give it as charity. As for the 'aqiqah (sacrifice for a newborn), it is not obligatory to give from it as charity or to eat from it; one may do with it as they wish, though it is preferable to treat it as one would treat the udhiyah. The ruling of the udhiyah applies to each sheep individually. Therefore, whoever slaughters multiple sacrifices must give a portion from each sacrifice as charity.