No, he didn't - the writers point that out in the commentary. Yancy Sr. tells Yancy Jr. that his son's name is Yancy, just like his name is Yancy, just like his grandfather's name is Yancy. He doesn't mention his father's name at all.
Yeah, you're right. Maybe Mildred's father's name was Yancy. That would be Yancy Sr.'s
grandfather, and the most likely explanation is that Mildred's last name is Fry, and her
father's name was Yancy Fry. I'm sure the writers were already planning the plot twist of "Roswell That Ends Well" when they wrote "Luck of the Fryrish". After all, they had "The Why of Fry" semi-planned at the time of "Space Pilot 3000", so Yancy mentioning his grandfather instead of his father was almost certainly written in order to keep these facts straight.
Anyway, it would look pretty screwy on a family tree: Phillip Fry II, son of Yancy Fry Jr., brother of Phillip Fry I, son of Yancy Fry Sr., son of Mildred Fry and Phillip Fry I. I wonder how the layout would look like?