For example, in my abstract algebra class, we recently proved that homomorphisms preserved identity; i.e., given groups $(G,*)$ and $(H,\#)$, if $\theta : G \rightarrow H$ is a homomorphism, then $\theta(e_g) = e_h$. This was the proof:
$$\theta(e_g) = \theta(e_g*e_g) = \theta(e_g)\#\theta(e_g) \implies \theta(e_g)^{-1} \# \theta(e_g)= e_h = \theta(e_g)$$
However, this fact seems obvious by the nature that $\theta$ preserves mappings, and $e_g$ and $e_h$ are determined by their mappings with other elements, under their respective operations.
Is there a mathematical way to talk to talk about groups in terms of the operation, instead of the set? Because elements are, in many senses, only meaningfully defined by their relationships with other elements (hence the power of isomorphisms).
I.e., is there a mathematical way to prove that the identity is mapped to identity under homomorphism without looking at a specific element? By saying something similar to, $\theta : * \rightarrow \# \implies \theta(e_*) = e_\#$ (because mappings are preserved)?
One way to approach this issue is by thinking about reducts. A reduct of a structure is a structure with the same underlying set but less, well, structure: we take the original structure and just "forget" some of it. (See also the term "forgetful functor," which I also mention below.)
The key reduction here is from groups to semigroups. Any group $\mathcal{G}=(G; *, ^{-1}, e)$ can be viewed as a semigroup (= a set with an associative binary operation) by "forgetting" about the additional structure: let $\mathcal{S}_\mathcal{G}=(G;*)$. Not every semigroup arises from a group this way, but every group gives rise to a semigroup in this manner.
The identity element and the inverse operation are then "redundant" in the following senses:
Suppose $\mathcal{G},\mathcal{H}$ are groups such that $\mathcal{S}_\mathcal{G}\cong\mathcal{S}_\mathcal{H}$. Then $\mathcal{G}\cong\mathcal{H}$.
Suppose $f:\mathcal{S}_\mathcal{G}\rightarrow\mathcal{S}_\mathcal{H}$ is a semigroup homomorphism. Then $f:\mathcal{G}\rightarrow\mathcal{H}$ is a group homomorphism. (There's a bit of abuse of notation going on here, but it should be clear what's going on.) This is essentially what you're talking about in the OP.
Now what's interesting is that these two observations "live on different levels:" the first follows entirely from the fact that $^{-1}$ and $e$ are definable from $*$ alone in a precise sense, whereas the second uses something specific about groups. To see what I mean about the second one, think of the situation with monoids. Again, every monoid yields a semigroup by "forgetting" the identity, but semigroup homomorphisms between semigroups arising from monoids are not homomorphisms on those monoids!
For example, let $M_2$ be the monoid with underlying set $\{0, 1\}$, binary operation $\max$ ($a \max b$ is just the maximum of $a$ and $b$), and identity element $0$, and consider the function $f:\{0, 1\}\rightarrow\{0, 1\}$ sending both $0$ and $1$ to $1$. The function $f$ is a semigroup homomorphism from the semigroup gotten from $M_2$ to itself, but is not a monoid homomorphism from $M_2$ to itself since it doesn't send the identity to the identity. The key algebraic fact here is that monoids aren't cancellative.
So we see that there are a few things at play here:
The idea of "forgetting" structure; keywords here are "forgetful functor" (from category theory) and "reduct" (from model theory).
The notion of "definability" (my first bulletpoint above; this is a key concept of model theory).
Algebraic properties of complicated structures which relate to morphisms between simpler types of structures (e.g. cancellation in the semigroups-from-groups versus semigroups-from-monoids point above; this is a key issue in universal algebra).
The three subjects that leap to mind as relevant here - besides abstract algebra, of course, which I think is a bit less focused on the particular issues you're interested in, and which I suspect you're already aware of - are model theory, category theory, and universal algebra. Frankly, I'm a model theory partisan: I think it probably matches nicely with the ideas you're having, and something like Hodges' textbook also talks about interesting algebraic issues with homomorphisms. But category theory and universal algebra also have lots to commend them, and in particular category theory is something you will definitely want to learn down the road.
It's worth noting that the idea of reducts does not take us away from talking about elements, and indeed the example above shows that groups form a special case in this setting. There are, however, some times when we can get away from talking about elements. These often crop up in universal algebra. For example, and close in spirit to what you describe above, given an algebraic structure (like a group) we can look at the "congruences" on the structure (think "quotient groups"); the congruences form a lattice, called the "congruence lattice," and we can work with this lattice as an algebraic structure on its own. Indeed, the congruence lattice forms an important object of study in universal algebra!
(It's also worth noting that category theory takes a similar-sounding attitude "one level higher:" it dispenses entirely with the "internal" structure of groups, and studies the category of groups - this is the class of all groups, together with group homomorphisms, the point being that (often) the important properties of groups are the ways in which they interact with each other rather than what they look like specifically. This is closely related to what you're talking about in spirit, but I don't think it's really the same thing; however, it's important and it seems rude to not mention it.)