What you say is close to a tautology. Ignoring the discussion of linear versus nonlinear, we can make a completely general statement about a system, which is that if the power input does not equal the power output, it must store or dissipate energy. So if power input does not equal power output, and you state that it does not convert the energy into other (externally visible) forms of energy, it must store that energy (note that the storage is a conversion to another form of energy, so the phrasing here is tricky, but I think I interpreted your meaning correctly).
So the remaining question is whether it is possible to create a non-linear system which conserves energy. The answer is trivially yes, because linearity or non-linearity does not speak to the meaning of the outputs, just that there's a non-linear response between them. It's just a mathematical statement, not necessarily a physical one. I could have a fixed force being applied over a horizontal distance as an input (the distance being the input variable), and my output is how far it pushes an object horizontally along a curve. It's easy to see in such a case that the energy conservation related the input work (force times distance) to potential energy (height of the output mass), but if the mass is being pushed along a curve, the horizontal distance wont scale linearly with vertical distance, so the output of the system (which I defined to be the horizontal position of the object), is nonlinear with respect to the input of the system (the distance the input force is applied over). Thus I can have a conservative system, which does not store any energy, but has a non-linear behavior simply because I chose the inputs and outputs correctly.
All linearity means is that if you feed two signals into the circuit $x_1(t)$ and $x_2(t)$, and observe the outputs $y_1(t)$ and $y_2(t)$, then that means that if you feed it the input $x_1(t) + x_2(t)$, the output will be $y_1(t) + y_2(t)$. And, slightly more generically, if you have any real constants $a$ and $b$, you can state that feeding it the input $a\cdot x_1(t) + b\cdot x_2(t)$ will result in the output $a\cdot y_1(t) + b\cdot y_2(t)$. Any system that is linear will have this relationship. Any system that does not is non-linear. That's all it means. There's a whole ton of neat convenient behaviors that follow for linear systems, but they all follow from this principle.
If you constrain the problem further to require the non-linear system's inputs and outputs to be measured in energy, then it must store or dissipate energy by tautology. If we have energy as an input and energy as an output, and the system does not store nor dissipate energy, then input power = output power must be necessarily true, by how we define what storing or dissipating energy means. $x=y$ is indeed a linear relationship, so a non-linear system whose inputs and outputs are measured in energy must store or dissipate because it would be linear if it didn't.
We can also have systems with multiple outputs. A cross-over network for speakers is such an example. A crossover network takes a full-spectrum audio signal from an amplifier and splits it between two speakers. It sends the high frequencies to one speaker and low frequencies to the other. Neither of the outputs need be linear with the input, but the combined energy output can, in the ideal limit, be the same as the input (in practical circuits, we do accept losses)
