AP style book doesn’t recognize it for non-medical doctors. Bill Cosby has a pHD in education. The only time anyone called him doctor was Dr. Huxtable.
GPO Style Manual, among others does include it. I don’t have access to to current AP but suspect it does—what you don’t mix is honorific and suffix. Anyway, it is simply a common courtesy and respect to include Dr. or Ph.D. And it is entirely appropriate to say Dr. Biden. Complain about her politics, about her husband, whatever, but let’s be respectful of the degree that she earned. I’m sure you would have no problem doing it if it was somebody with politics you agreed with. So do it here.