Is a doctor who goes into Internal Medicine specializing in Infectious disease still consider to be a primary care doctor?

Usually not. He/she is certainly qualified since they did an internal medicine residency. But since they specialized in ID they most likely limit their practice to ID.
Most . Who specialize in infectious disease focus on this. But they can also be a primary care doctor. So It really depends on the doctors focus in practice.
Primary care. General internal medicine is one of several primary care specialties. An infectious disease specialist is an internist who has done a fellowship in that sub specialty and usually practices as a consultant, but may also continue to practice general internal medicine.