It flows, you use fluid equations to describe it's state. I will grant that it flows in a different medium than molecular fluids. but that does not prevent it from being a fluid.
As for what is flowing, my understanding is that it is electrons in an em field. which if you think about it is almost the same thing that is flowing in a molecular fluid, only that has drag a bunch of protons around at the same time limiting performance.
The fluid analogy is ok for introducing the concept of electricity but it has some significant limitations, so one should grow out of it fairly quickly if you want a deeper understanding of the subject.
As for what is flowing, my understanding is that it is electrons in an em field. which if you think about it is almost the same thing that is flowing in a molecular fluid, only that has drag a bunch of protons around at the same time limiting performance.