Can a non-normal distribution have the same mean and median? This question seems to be related to: Does a "Normal Distribution" need to have mean=median=mode?

Yes.
The simplest way for a distribution to have the same mean and median is to be symmetric. There's a huge range of symmetric distributions, from Bernoulli (two points) to uniform to $t$-distributions, to multimodal distributions.
But symmetry isn't needed. You can take any distribution and make small changes to it to get the mean and median to be equal. If the mean is less than the median, you can increase the mean by adding a small probability of a very large value. If the mean is greater than the median, you can decrease the mean by adding a small probability of a large negative value.
