Moore's law deals with the number of transistors in an integrated circuit, which is used as a proxy for computational power. In a quantum computing device the analogy would be the number of qubits. However, this by itself would be a poor benchmark, namely because it is easy to build lots of qubits. Building many qubits with properties such as long coherence times, low crosstalk, good readout fidelity etc... is what is actually difficult to achieve. So in the quantum world, number of qubits is not a proxy for computational power. Instead, it is the number of number of qubits combined with the quality of those qubits, as measured by their error rates and coherence times, that determines the overall computational power of a quantum device.
The notion of "computational power" is the next question. To quantify this we need some kind of benchmark that we can use to quantify the performance of a give quantum device, and compare it to others. One such benchmark for gate-based quantum devices is the Quantum Volume, developed by IBM  (disclosure, I am on this paper). This is a benchmark using random quantum circuits to mimic the performance of an average quantum circuit on a given device. It takes into account both the number of qubits and their quality as is necessary for quantum devices (at least noisy ones). IBM uses this as their benchmark and have seen exponential scaling of this value every year; they aim to double the Quantum Volume every year. There are of course infinitely many benchmarks possible, and a suite of benchmarks is likely at some point in the future, however Quantum Volume seems to be gaining traction. See for example Refs.[3,4].
Note also that the 53-qubit Rochester device from IBM has been out for some time, and can be accessed by members of the IBM Quantum Network. All of the devices, along with their Quantum Volume, can be see on the IBM Quantum Experience website .