Dentists ARE doctors but the cause for confusion has more to do with how people define and perceive what being a doctor means. A doctor is actually a title and not a profession.
However, due to cultural norms people have often associated the title doctor with being a physician. This perception is the sole cause for the confusion but we will shed clarity on this matter to end it once and for all.
A doctor is a title
A doctor is an academic title, which is usually bestowed upon those who've reached the pinnacle of their subject of study. What this means is that there are a lot of different doctors. In fact, most fields have their own respective academic degree which grants a doctorate, the title of doctor.
Examples of doctorates:
MD - Medical Doctor
DO - Doctor of Osteopathic Medicine
DDS - Doctor of Dental Surgery
DMD - Doctor of Dental Medicine
OD - Doctor of Optometry
DPT - Doctor of Physical Therapy
PhD - Doctor of Philosophy
We would like to emphasize the PhD degree because a LOT of subject disciplines offer it. Basically any subject that you can think of will most likely have a PhD degree. It is all inclusive. As long as you complete all the requirements and learn everything that you could possibly learn about your subject, you'll be awarded the PhD. Although it is often a long and arduous task that takes most people many years to complete.
A PhD differs from the other degrees listed above because it does not specify what the individual is an expert in. The other degrees are very explicit in what their doctorate is in such as MD, which is a medical doctor. They are the experts in the field of medicine. A PhD could mean anything from chemistry to English literature. Therefore, people usually have to add additional info in order to clarify what their PhD is in.
Do dentists have the title doctor?
All dentists hold the title doctor because their degree grants a doctorate. In the United States, all dental schools either grant a DDS (Doctor of Dental Surgery) or a DMD (Doctor of Dental medicine). They're both equivalent but the difference depends on the state which grants the degree.
For instance, all dental schools in New York grants the DDS degree while schools in Massachusetts grants the DMD degree. The state degrees which degree and all schools within the state have to follow suit. This means that two schools in the same state cannot offer a differing degree.
Where did the idea that all doctors are physicians come from?
Whenever you go to a healthcare clinic or a hospital, the healthcare provider that comes in to treat you will introduce themselves as Doctor so and so. You will most likely never hear them say, "My name is XYZ and I am a physician." No, they will always keep it concise by stating that they are Doctor XYZ. That establishes authority and legitimacy within a few seconds.
Sometimes, people have some very difficult last names and you may not have remembered what they introduced themselves as because you were highly concerned about the medical condition that you have. You were intently listening in on what they were saying about your health but seem to have forgotten their name. All you remember is that they were Doctor something. Therefore to make it accurate and not offend the other person, you just call them "Doctor" and not even mention their last name!
Aside from getting your health evaluated, you don't really encounter too many other instances where the person you are interacting with has a title. Therefore, for simplicity's sake, whenever people talk about going to the doctor what they are implying is that they're going to see their physician.
Overtime, that habit has just become entrenched within our minds and we just started associating doctor with physician.
All dentists are doctors because a doctor is a title and does not precisely describe a profession. A medical doctor is also a doctor but their expertise is in medicine and they are physicians.
Basically all dentists are doctors but not all doctors are dentists. The same could be said for physicians. All physicians are doctors but not all doctors are physicians.
However, just let it be known that when someone says that they're going to the doctor, they're most likely referring to their physician. That means they have a medical doctor appointment.
We should still understand context and cultural norms and that is most likely what the general population refers to when they make that statement.