Also, for the camp of people there who say it's just differential equations and see the right side of the above equation. I think they've not seen circuits enough. There are very many cases where the circuit itself offers quick ways to arrive at simplified equations.
The section I bolded out of your question (actually, and edit to your question) is probably the most useful for understanding the variety of answers you are receiving. You ask "What exactly are electrical circuits as mathematical objects?" On its own, this is a easy question to answer in the negative. Electrical circuits are not mathematical objects. They're physical objects. But most likely what you're really asking is what mathematical objects are used to model electrical circuits. And when we come to modeling, it's all about intended use. A model cannot be applied meaningfully to a real system unless one considers the intent behind using the model.
Some would like a model which truly captures the behavior of any electronic circuit. For those people, Maxwell's Equations are the definitive model of electronic circuits. And I use "definitive" properly here -- we define electronic circuits to be a subset of the things governed by Maxwell's Equations. So in that sense, differential equations over a field is the best mathematical object to model a circuit.
But "there are very much cases where the circuit in itself gives quick ways to arrive at simplified equations." This is absolutely true. In many cases, we are comfortable assuming some details don't matter. In the hierarchy of mathematical objects used to model electronic systems, the "distributed element model" is the next most common model. In this model, we assume that many of the interactions between parts of the field do not matter. In particular, we assume that electrical fields closely follow conductors, and magnetic fields do not "leak." For instance, if we have two inductors side by side, Maxwell's equations says that they will magnetically couple, affecting each other. But, in practice, we can often design circuits where this effect is negligible.
This is the first model where Kirchoff's laws apply, because we can speak to "wires" as individual discrete entities, rather than regions with a given permittivity and permeability. This is the first point where a "graph" is a meaningful model. However, it is an infinite graph. When modeling systems as distributed elements, we come across interesting concepts like transmission lines, which are modeled as an infinite series of infinitesimal inductors, resistors and capacitors, and we apply calculus to it.
The next most common big leap in modeling assumptions is to assume that some properties are evenly distributed across regions. For example, we can assume that the capacitance of all points in a capacitor can be treated as "the same." This is the lumped element model., another very well accepted way of modeling electronic circuits. Once we accept these assumptions we can develop the finite graphs that you are most likely familiar with, which are solved directly via KVL and KCL (rather than having to apply those laws over infinite sums using calculus).
Beyond this, many circuits admit even more useful models. Consider linear circuits. Not all circuits are linear -- circuits with transistors in particular are famously non-linear. But those using just resistors, capacitors, and inductors are linear. For these, the most useful models are often the Laplace transformation of the circuit's behavior. The Laplace transform has the wonderful property of converting all linear systems (including linear differential equations) into algebraic objects. If you work in control theory, these are the bees' knees because it is so much easier to work with algebra than differential equations!
And some non-linear circuits admit convenient mathematical models too. By far the most famous is that of saturated transistors, which form the basis of every computer you've ever seen. These are extremely well modeled by boolean logic. I find this interesting because it shows that this series of models isn't even a simple progression from complex to simple. There are many branches of intended uses for electronic circuits, each of which has their own mathematical objects which are most convenient for modeling.
And sometimes its not even the circuit that matters, but your intended use for the circuit. Amplifiers involve decidedly non-linear circuits (large transistors), but we drive them in such a way that we find a "linearized" model is applicable. This is the so called small-signal model, which does not fully characterize the circuit's behavior, but it does characterize the part of the behavior we care about (the amplification of an input signal).
So every one of these models is applied to electronic circuits on a regular basis. Which model one uses depends on the intended use of the model, and the assumptions one is willing to bake in. One will find this is true for all modeling, not just that of circuits. So pick your mathematical objects wisely. Consider the intended use of the model, and go from there.
