> whether this situation could fall under fair use with respect to the person recording?

IMHO the person posting the YouTube video with the Taylor Swift song in background would have an excellent shot at a fair-use defense against any claim of infringement of the copyright in the song or the recording (which could be two different things). See, e.g., https://fairuse.stanford.edu/overview/fair-use/four-factors/

I also strongly suspect that the owner of the copyright(s) in question would think very hard before making an infringement claim in the first place, for fear of the adverse publicity.