Based on the passage in Shmot 12:4, the animal brought as the korban Pesah can be either a goat or a lamb.
Mishna: In the case of one who says to his slave: Go and slaughter the Paschal offering on my behalf, but does not specify which type of animal to slaughter, the halakha is as follows: If the slave slaughtered a kid, his master may eat it; if he slaughtered a lamb, his master may eat it. If the slave slaughtered both a kid and a lamb, his master should eat from the first one that was slaughtered; the second is invalid and should be burned.
The Gemara notes that the first rule in the Mishna (when the master does not specify which type of animal to bring) must refer to a case where the servant chose the animal not usually favored by his master. The Mishna teaches that, nevertheless, we assume that the master gave free reign to the servant in choosing which type of animal to prepare.
With regard to the second rule, the Gemara asks how the first one can count, when we know that a person cannot be counted on two sacrifices, and yet the servant had slaughtered both on his master’s behalf!? The Gemara answers that the Mishna must be talking about a very specific case: the case of a king and queen. To support this contention, the Gemara tells a story about a king and queen who ordered their servant to prepare a korban Pesah. The servant slaughtered two animals. The king deferred to the queen, who deferred to Rabban Gamliel. Rabban Gamliel ruled that although the law usually is that, in such a case, neither animal can be used as the Pesah, in the case of the king and queen who aren’t so concerned about an animal going to waste, the first one would be used for the sacrifice.
Maimonides explains that this unique ruling for royalty is based on the concept of shalom malkhut – seeking peace with the monarchy – which is explained either as a concern lest the king and queen become angry with the Sages, or that they become angry with their servant, leading to a severe punishment and even death. The Tosafot Yom Tov explains that this story is brought as an example of how the king and queen relied totally on the rulings of the Sages, therefore the Sages could make a decision on their behalf and they would not be considered signed up for two korbanot.
The story in the Gemara seems to be referring to the period of Rabban Gamliel ha-zaken, the grandson of Hillel, making the king Agrippas I. This is one of many stories that appear in the Gemara extolling King Agrippas’ respect for the words of the Sages.