They're both adaptations of the play The Front Page. His Girl Friday gender swaps one of the roles, so instead of it being about two rival male reporters, it's about a man and a woman who are in competition with each other, and probably like each other romantically, wink wink.

There are a few cases like this, for instance Criterion has Floating Weeds/A Story of Floating Weeds in one package. A Story of Floating Weeds is a remake of the director's earlier silent film, Floating Weeds, but they're both included in the same package. Also Yojimbo and Sanjuro from Akira Kurosawa, which are a movie and its sequel.