This feels like the right-wing equivalent of pronoun policing lol.

Nothing about it feels "right-wing" at all -- actual conservatism is all about maintaining continuity and avoiding forced changes to traditional conventions and established social norms.

Trump and his followers are a competing faction of the extreme left, as should be readily evidenced by his economic policies, and it's not at all surprising that he shares the same mentality of inserting political shibboleths into mundane aspects of everyday life.