Let's say, a generic consumer-level CRT. How much colors can it reproduce?
"Trick" question: it's an analogue device, so it is not limited to any particular fixed number of colours and can continuously vary between those colours in its gamut.
Analogue colour depth is probably best considered in terms of signal-to-noise ratio or contrast ratio.
Edit: you also need to consider how well the colours correspond to the source, see "Color rendering index".