Arizona State has a Phd in Design - not ID specifically.
It’s not so straightforward to define “PhD in ID” since at such level of specilization you should consider most “PhD in Design” programs and then match your specific topic of interest with that of existing Professors or Supervisors.
i know the royal college of art does, too
Carnegie Mellon University, also.
Unless you want to be a college Dean or museum curator, skip the Phd.
it doesn’t equal talent.