Yes, its nicknamed after him because he was the president of the organization now known as the MPAA when they developed the Motion Picture Production Code. That he was also a Republican politician is mostly beside the point; the modern left-right alignment of the parties and the particular association of Republicans with the religious and “family” groups to whose concerns the code was a response wasn't a thing in the 1930s.

Hays was a republican. And also president of MPAA. That one republican worked towards the Hays code is not evidence that the GOP as a whole desired it.

Personally, it seems to me to be a code developed at a time of differing cultural values, that likely would have been supported by any mainstream politician.