Is it permissible to share in a single sacrificial animal on Eid al-Adha with different intentions, such as one partner intending it as an Eid sacrifice (udhiyah) and the other intending it as an 'aqiqah (sacrifice for a newborn)?

Aqiqah is a Sunnah, not an obligation. Its origin is the Prophet's (peace be upon him) saying: "With the boy is an Aqiqah, so shed blood for him and remove the harm from him." It consists of two sheep for a male and one sheep for a female, based on the Prophet's (peace be upon him) saying: "For the boy are two equivalent sheep, and for the girl is one sheep." According to the Shafi'i school, it is permissible to slaughter for Aqiqah what suffices for Udhiyah (a complete sheep or one-seventh of a camel or a cow), and it is preferable to perform the Aqiqah for the newborn separately.