Note: since no location/culture/company policy is provided in the question, no guarantees can be made. However, your interpretation of what constitutes sick leave seems to diverge from the general interpretation, which is what this answer is responding to.
One could take an hour or two off work to go to the doctor and then continue working.
However, in my case, my doctor is located in another city. Therefore, it will take around 6 hours to go, have the appointment, and come back.
The length of time you are absent does not define whether it is sick leave or not. Sick leave is defined by the nature of the absence, not its length.
There seems to be a misunderstanding on your part on what sick leave constitutes. Sick leave is not "medical appointment leave". Sick leave is granted when unable to work due to illness. Illness often entails medical appointments, but medical appointments do not always entail illness.
When you are ill and take sick leave, you generally make an appointment with a medical professional, but that does not mean that every appointment with a medical professional therefore entails sick leave. A very clear cut example of this distinction would be elective plastic surgery.
That being said, "sick leave can't be planned" is also an oversimplification on your employer's part. This can range from dentist appointments (which even in urgent situations often need to be planned one or two days ahead) to treatment for illnesses which don't compromise your ability to work on a daily basis unless you miss regular treatment.
While I suspect that your employer may have overstated their case by stating that there's no such thing as planned sick leave, it's possible that their basis for rejecting your sick leave application is valid.
That being said, without specifying a location, no final conclusion can be made on the legalities of this situation.
I am not asking for any information regarding my specific country/company policy. I am just asking if "planned sick leave" is a thing.
You seem to think that the definition and workings of "sick leave" are universally defined. They are not. There is no legal definition that transcends national borders. The largest (currently existing) legislative scope is a country's legal system.
It's impossible to fully answer the question without knowing the country in question.