However, R2-D2's memory is not wiped; as a result, R2-D2 is the only surviving character at the end of the saga who knows the entire story of the Skywalker family. Both R2-D2 and C-3PO end up in the possession of Captain Raymus Antilles (Rohan Nichol) onboard the Tantive IV.

Why didn't R2 have his memory wiped?

A memory wipe was a method that was used to selectively or completely erase the memory of a droid. After Cad Bane had kidnapped and ransacked the memory of C-3PO and R2-D2 for information on the Senate Building's layout, he only had the incident wiped from their memories to avoid having any suspicion falling on him.

Does R2 remember Anakin?

Because Anakin never wiped his memories, and Bail Organa didn't wipe it at the end of Revenge of the Sith, R2-D2 is the only character who remembers everything from the first six movies. He was also there for every major moment. While it might seem like a coincidence, it's not.