Channel Capacity - is this only for discrete/digital signals?
The definition of channel capacity can be applied to either digital or analog cases. The meaning depends on how you calculate it. The definition is:

$$C=\max_{p_X(x)}\ I(X;Y)$$

In the digital world, $p_X(x)$ is a mass probability function and the mutual information is calculated as $H(Y)-H(Y|X)$, where $H(\cdot)$ denotes the entropy.

In the analog case, then $p_X(x)$ is a probability density function and the mutual information is calculated as $h(Y)-h(Y|X)$, where $h(\cdot)$ denotes the differential entropy.

Regarding your other question, channel capacity and bandwidth are not the same thing. Take for example a Gaussian channel, where $$C=B\log_2(1+\mathrm{SNR})$$

You can see that the capacity depends on the bandwidth, but they are clearly not the same.