In physics I always see formulas with variables representing properties of objects like (this is a simplified example):
F = a bodies force.
m = a bodies mass.
a = a bodies acceleration.
But why do I never see variables representing whole objects like
B = a body
and is there actually any way of doing that?
The variables in physics eventually represent things that are experimentally measurable. That is quantities like $9.8 \mathrm{\ m/s^2}$, or even dimensionless ratios of such quantities. It is physically meaningful for one such quantity to be twice as large as another, or to perform mathematical operations on such quantities. So it makes sense to put them in mathematical formulas.
For $B= a \mathrm{\ body}$, what would it mean to have $B^2$ or $-B$? There doesn’t seem to be much in the way of sensible mathematical operations you could perform on such a quantity. So we would not see it in mathematical formulas.
It really isn't the way things are done. $F$, $m$, and $a$ are things that can be measured. $F = ma$ is a relationship between those measured values.
You might have a collection of bodies that all exert forces on each other. You might name them object 1, object 2, etc. One example is each object is an atom in a block.
Newton's laws tell us that every action has an equal and opposite reaction. This means that if object 3 exerts a force on object 7, object 7 also exerts a force of equal strength and opposite direction on object 3. We might represent this with notation like $F_{3,7} for the force on object 3 from object 7.
$$F_{3,7} = -F_{7,3}$$
or more generally
$$F_{n,m} = -F_{m,n}$$
