First, the cable is no "wire" it is a TEM -mode waveguide; the coaxial structure supports a wide-band spectrum of many signals.
Due to the fact that at any moment there are numerous signals propagating in cable-TV cable, it requires a spectrum analyzer to measure each signal power separately; the "noise" is only found in gaps between signals on the frequency scale.
In cable TV, the signal-to-noise ratio is rather named signal-to (noise plus interference) ratio. A TV tuner and receiver must utilize filters to separate desired signals from the undesired noise and interference (any signal not allowed to pass to receiver demodulator).
In digital cable TV, signal processing is different from the "analog" cable system; in addition to filtering, the desired signal is processed by digital signal processor which allows a higher level of noise and interference at the input from the cable. In digital systems, the ratio of pulse energy-to-noise (Eb/No) is used at signal input while BER, bit-error-rate at the output is the overall measure of demodulated and processed signal quality.