Since force is a vector, three components are enough to define it completely at some point in space and instant of time. Yet the electromagnetic force requires two independent forces, the electric and magnetic, and therefore six components to fully define it.
What is it about the electromagnetic force that requires six components to completely define it, rather than just three for force generally?
You actually only need three fields to define the electromagnetic force. What you're calling "numbers" are actually fields - independent numbers at every point in time and space.
Now, why does it take six numbers to nail down the electric and magnetic fields? Well, you can take two approaches. First, the definition of the electric field is you take a small test charge, place it at some point, and then measure the force on that test charge without letting it move. Divide that force by the charge on the test charge, and that gives the electric field. Because force is a vector, that needs three components to describe it, and if you do it everywhere that defines the field.
To measure the magnetic field, first measure the electric field. Then measure the force on a moving test charge. If you subtract from the measured net force the force applied by the electric field, the remaining force is magnetic. Defining the magnetic field requires multiple test charge measurements, since the velocity of the test charge figures into the calculation.
The content of the above two paragraphs is given by the Lorentz force law $$\vec{F} = q\vec{E} + q\vec{v}\times\vec{B}.$$ Note that the magnetic force, the second term, must always be perpendicular to the velocity. If we ever find that the difference between the net force and the electric force on a point charge is not perpendicular to it, then there is some other force we failed to take into account. One such possible error that would ruin our estimate of the electric field would be failing to take gravity into account, though that can be worked out by using test charges of different masses (e.g. protons and neutrons), or using uncharged masses to measure $\vec{g}$.
A deeper understanding comes from examining these quantities in special relativity. In that formalism, the position is given by what's called a 4-vector, often written as $x^\mu$ (the Greek superscript, $\mu$, goes from 0 through 4), collects position and time of events as $(x^0,x^1,x^2,x^3) = (ct,x, y, z)$. If we define a quantity along the path a charge takes through spacetime called the proper time $$\tau = \sqrt{(ct)^2 - x^2 -y^2 - z^2} = \sqrt{(x^0)^2 - (x^1)^2 - (x^2)^2 - (x^3)^2},$$ then we can take the derivative of each component of the 4-vector position along the path to get something called the 4-velocity $$v^\mu = \frac{\operatorname{d} x^\mu}{\operatorname{d}\tau}.$$ The usefulness of doing this is that we can write the Lorentz force law as $$m\frac{\operatorname{d} v^\mu}{\operatorname{d} \tau} = \sum_{\nu=0}^4 F^\mu_{\hphantom{\mu}\nu} \, qv^\nu,$$ where we have collected the components of the electric and magnetic field into a matrix (tensor) called the electromagnetic tensor. Why that tensor is antisymmetric, I'm not going to address here. If you count the number of independent components in an antisymmetric $4\times4$ matrix, you'll find there are six.
The thing is, those six fields are not actually all independent. We can reduce that to four fields using some vector calculus identities to satisfy half of the Maxwell's equations (the ones that don't have charges/currents in them - called "homogeneous") automatically. We define the electric potential, $\Phi$, and magnetic vector potential, $\vec{A}$, to be the fields that give us: \begin{align} \vec{E} & = -\vec{\nabla} \Phi - \frac{\partial \vec{A}}{\partial t} \\ \vec{B} & = \vec{\nabla}\times \vec{A}. \end{align} That cuts it down to 4 fields.
We can do one better, though. There are changes we can make to $\Phi$ and $\vec{A}$ that will give us the same electric and magnetic fields. Making a change like that is known as a gauge transformation. For any single field $\Lambda(t,\vec{x})$ we can change $\Phi$ and $\vec{A}$ by \begin{align} \Phi \rightarrow \Phi - \frac{\partial \Lambda}{\partial t} \\ \vec{A}\rightarrow \vec{A} + \vec{\nabla}\Lambda \end{align} and the new $\Phi$ and $\vec{A}$ will produce the same electric and magnetic fields, no matter what (smooth) function we use for $\Lambda$. That means that there are really only 3 independent fields that define the electric and magnetic fields/forces, even though there are four components in what is called the 4-potential $A^\mu = (\Phi/c,\vec{A})$.
Why does it work out there are only three independent components to the electromagnetic field? The reason, ultimately, is tied to the fact that charge is conserved. Look at the Lorentz force law, again. That says that the force is fixed by the interaction of the electromagnetic field with the four-velocity of the charged particle. Well, that combination $q v^\mu$ is an example of something called the 4-current that collects charge density and current density as $J^\mu = (c\rho, \vec{J})$. Because charge is conserved, the 4-current has to satisfy $$ \frac{\partial \rho}{\partial t} = -\vec{\nabla}\cdot \vec{J}. $$ That condition means that the 4-current density has only three independent components, everywhere, and thus the electromagnetic field has only three independent fields that fix it (see also: the other two Maxwell's equations [called inhomogeneous]).
The electromagnetic force is regarded as one unified field because that is what it is in the 4D spacetime formalism of Minkowski: it is not a vector field but an antisymmetric 4x4 matrix field. Being antisymmetric, there is no time/time component nor diagonal space/space components; the three independent off-diagonal space/space components form the "magnetic field" and the three independent time/space components form the electric field: $$F_{\downarrow\rightarrow} = \begin{bmatrix} 0&E_x&E_y&E_z\\ -E_x&0&-B_z&B_y\\ -E_y&B_z&0&-B_x\\ -E_z&-B_y&B_x&0 \end{bmatrix}$$ But in the form that it is "electromagnetic" it is not a single force field; it is a rank-2 tensor (matrix) field; in the form that it makes rank-1 tensor (vector) fields, it has to be described with two of them, an "electric force field" and a "magnetic force field."
