Channel Capacity - is this only for discrete/digital signals? The units of channel capacity is bits/second. Does this mean this only refers to discrete/digital signals? Is channel capacity analogous to bandwidth in case of an analog signal transmission?
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.