The later society may not know about the accomplishments of the earlier society.
Possible but to my mind unlikely.
Such an undertaking would be a major project that would consume a lot of assets. Particularly if it was a Generation Ship.
Such a mission would of historical significance on parallel with the campaign to reach the moon.
Maybe the moon landing will slip in to the obscure facts of history in the; future who can tell. Perhaps it will if trips to the moon become as common as trips to the Grand Canyon.
But I just don’t think it would. If a mission is planned that will duplicate a mission that set out 125 years earlier but had not been completed yet and the new mission will make the trip in 2 years I am fairly confident that some history buff would bring it to the attention of some news writer and it would get public attention.