I was wondering if it is possible to consider a category $\mathcal{C}$ as a variety of algebras. My idea was to consider pairs of composable morphisms of $\mathcal{C}$ as algebras, compositions and identities as binary and nullary terms satisfying associativity and neutrality axioms. Is that possible or not? And if it's possible, is it true that $\mathcal{C}$ is a variety? In that case, what would be the set of axioms $\mathcal{C}$ is a model for?
Problem is that I don't know how to make a variety of algebras out of a category, mine was only a rough and clearly wrong attempt, as pointed out in comments. My question was: is there somebody who tried to put a structure of a class of algebras on a category? For examples, fields cannot be expressed as algebras of any type becuase of the impossibility of dividing by zero. What would be the obstruction(s) in case of categories?
I think there's a bit of confusion here, in particular between algebras (which are individual structures) and varieties (which are collections of algebras). This is reflected in your question (where morphisms are treated both as structures (or rather, "halves" of structures) and as elements of an algebra) and your statement
which is not true - fields are indeed algebras. What is true is that the class of fields is not a variety.
So I think there are really two questions here:
Categories-as-algebras: Is the class of categories somehow a variety? For this version of the question, let's focus on small categories for simplicity.
Categories-as-varieties: Is there a natural way to identify an arbitrary category with a variety? If not, when can we do this?
The first question is pretty easy to answer. While a category isn't an algebra in the usual sense, since composition isn't total, it is a "partial algebra." The axioms for categories are then equational in the sense of partial algebras, and so the class of (small) categories does indeed form a "variety" of partial algebras.
The second question is more complicated, since there are in principle many, many ways to "interpret" a category. (Also, since nontrivial varieties are proper classes we now need to shift to large categories.) Let me make a couple observations:
Right at the start, the most natural thing to try would be to identify objects in the category with algebras. Even this breaks down immediately, since there are categories which are not concrete, that is, can't be interpreted as categories of sets with structure in any way.
While a nontrivial variety $\mathcal{V}$ can be interpreted as a large category $\mathcal{C}_\mathcal{V}$ - where the objects of $\mathcal{C}_\mathcal{V}$ are the algebras in $\mathcal{V}$ and the morphisms in $\mathcal{C}_\mathcal{V}$ are the homomorphisms between algebras in $\mathcal{V}$ - there is no reason to expect the converse to hold, even for concrete categories. E.g. any category coming from a variety in this way must have products, and not all categories do. So in general this isn't going to work.
Finally, let me mention a lovely result, not quite what you're looking for but still in a related spirit (I think): the Freyd-Mitchell embedding theorem. This theorem states roughly that any small abelian category can be viewed as a category of modules over some ring $R$ in a particularly nice way. This is very far from what you're asking, but does to me have a similar flavor, and the theorem has some interactions with universal algebra.
CODA: Algebras-of-operators-on-varieties-of-algebras (don't let this confuse you - read the rest of the answer first!)
By Birkhoff's HSP theorem, varieties are exactly those classes which are closed under taking arbitrary homomorphic images, substructures, and (arbitrary-arity) products. This suggests looking at the semigroup (sometimes, monoid) generated by elements $\mathbb{H},\mathbb{S},\mathbb{P}$, thought of as corresponding to the closure operations "add all products," "add all homomorphic images," and "add all products" respectively - with the semigroup operation viewed as composition of operators and equalities coming from "equality-on-all-classes-of-algebras." This monoid satisfies a number of nontrivial equations:
$\mathbb{X}\mathbb{HSP}(x)=\mathbb{HSP}(x)$, for $\mathbb{X}\in\{\mathbb{H, S, P}\}$, since $\mathbb{HSP}(x)$ is already a variety regardless of what $x$ is.
$\mathbb{SHS}(x)=\mathbb{HS}(x)$ (exercise).
And others.
This gives us
an equational theory
which describes a "structure" whose elements are operations
which act on possibly proper classes
whose elements are themselves algebras.
Whew!
(This theory does, of course, have set-sized models though - so it also corresponds to a variety in the usual sense.)
Things get even more interesting when we further restrict attention. E.g. if we look only at classes of algebras of a particular type, more equations might be satisfied - e.g. when we look at classes of Boolean algebras, we have $\mathbb{HS}=\mathbb{SH}$ even though this is not true in general. Conversely, some non-identities continue to hold (e.g. for metabelian groups we still have $\mathbb{HSP}\not=\mathbb{SHPS}$. We can also study other operators on classes of algebras, like closure under finite products or elementary substructures (the former has been intensely studied; I'm not sure about the latter, but it seems interesting to me).