The Maharal commenting on Rashi, answers that
this was the way the people back then made Shevu’ot; the one swearing would place their hands under the other thigh of the person he is swearing to (as the Ibn Ezra points out in his commentary to Breishit 24:2 and confirmed by Da’at Mikra ad. loc.). Yaakov thought that if he did not do the Shevuah this way, when Yosef would ask him to go and bury his father, Paroh would respond that he did not make the Shevuah according to the law of the land, and, therefore, the Shevuah was null and void.
The Maharal continues to say that the reason Rashi gave the explanation by Avraham that the bris was his nekitas cheifetz is because it wouldn't make sense to say that Avraham was following the world's custom. Avraham had no reason to have Eliezer swear like the custom, unlike Yaakov, who was concerned that Pharaoh wouldn't recognize anything else.
The ibn Ezra himself would probably give the same explanation for both Avraham and Yaakov.
Much thanks to @b a for helping me understand the Maharal correctly.