Why are the electric force and magnetic force classified as electromagnetism?
Consider this: A charged particle at rest creates an electric field, but no magnetic field. Now if you walk past the charge, it will be in motion from your point of view, that is, in your frame of reference. So your magnetometer will detect a magnetic field.
But the charge is just sitting on the table. Nothing about the charge has changed.
Evidently the space around the charge is filled with something that at times appears to be a pure electric field, and at other times appears to have a magnetic field. We conclude that the field is something other than an electric field or a magnetic field. It is another type of field which combines the two into one entity.
The arguments from special relativity given in the other answers is correct. What is charge according to one observer is current according to another observer that is in relative motion to the first. But this is, from a historical perspective, somewhat backwards. This consideration is what led Einstein to develop special relativity -- the paper is called On the Electrodynamics of Moving Bodies. But long before Einstein, it was known that electricity and magnetism were not independent.
Have you seen Faraday's law of induction? If a magnet is moved through a coil of wire, there will be an induced electric field in the wire and a measurable current produced as a result. This is a hint that electricity and magnetism are related.
It also seems that you haven't seen Maxwell's correction term to Ampere's law. Ampere's law with only the current as source is not consistent with the conservation of electric charge. For Ampere's law to be consistent with conservation of electric charge, one must include a source term proportional to the rate of change [time derivative] of the electric field. Again the electric and magnetic fields are interrelated.
It is not possible to formulate satisfactory, independent theories of the electric and magnetic field. They will always be coupled. From a modern point of view, it has to be like this because there is no way to be consistent with Einstein otherwise, but it was because of Maxwell's theory of electromagnetism that special relativity was developed, not because of special relativity that it was realized that electricity and magnetism are related.
You are right, the electric field and the magnetic field are distinct fields that have different properties. The reason why they are still classified as the cause for the "electromagnetic force" are the following:
In higher theories, like the field theory, the electric and the magnetic field are caused by the same gauge principles. There is just "one" interaction between a particle and the electromagnetic field, and this electric field and the magnetic field are two fields to describe this interaction.
Furthermore: If you change your frame of reference, for example by running by the experiment you are looking at at the moment, then electric field and magnetic field will mix up. What part of a force acting on a particle is magnetic and what is electric is a property that depends on your relative speed towards the particle. Here you see again, that electric field and magnetic field are just two ways to describe the "electromagnetic interaction". How this interaction splits up to the E-field and the B-field depends on your frame of reference.