Voltage \(V\) is standardized in terms of the Josephson effect. In a superconductor, electrons pair up and condense into a quantum state that is described by a complex number called the order parameter, with a magnitude and a phase. The magnitude is related to the density of pairs. The phase is related to the coherent response of all the pairs, and only takes on a well-defined value below the superconducting transition. In a junction between superconductors (say a thin tunneling barrier of insulator), a dc voltage difference between the two sides causes the phase to "wind" as a function of time, leading to an ac current with a frequency of \(2eV/h\). Alternately, applying an ac voltage of known frequency \(f\) can generate a dc voltage at integer multiples of \(h f/2e\). The superconducting phase is an emergent quantity, well defined only when the number of pairs is large.

The Ohm \(\Omega\) is standardized in terms of the integer quantum Hall effect. Electrons confined to a relatively clean 2D layer and placed in a large magnetic field show plateaus in the Hall resistance, the relationship between longitudinal current and transverse voltage, at integer multiples of \(e^{2}/h\). The reason for picking out those particular values is deeply connected to topology, and is independent of the details of the material system. You can see the integer QHE in many systems, one reason why it's good to use as a standard. The existence of the plateaus, and therefore really accurate quantization, in actual measurements of the Hall conductance requires disorder. Precise Hall quantization is likewise also an emergent phenomenon.

Interesting that the fundamental definition of the kilogram is deeply connected to two experimental phenomena that are only quantized to high precision because they emerge in condensed matter.