A histogram approximation lets you approximate any arbitrary distribution by slicing it up into little bins and computing the probability of seeing data in a bin. It doesn’t make any assumptions about the shape of the distribution, which is the difference between it and the MvN approximation.
I guess you could use it on data that is already discrete? But I also have no idea, I’ve never done anything like that.
I’d recommend starting with MvN though, as all well-behaved posteriors should be converging to something normalish anyway.