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This question seems to be related to: Does a "Normal Distribution" need to have mean=median=mode?

Jantje Houten
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    Take a Weibull distribution with shape parameter $3.44$: It is slightly right-skewed but the median and mean are identical. – COOLSerdash Aug 16 '21 at 08:18
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    There are many posts on site which answer this question – Glen_b Aug 16 '21 at 09:42
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    e.g. https://stats.stackexchange.com/questions/125084/does-mean-median-imply-that-a-unimodal-distribution-is-symmetric points to several examples of asymmetric distributions for which it is the case. – Glen_b Aug 16 '21 at 09:48

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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.

Thomas Lumley
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    It is easy to find well-behaved asymmetric distributions in which the mean is equal to the median. The binomial with probability of success 0.2 in 5 trials has mean and median 1. 0,0,1,1,1,1,3 is a not very magnificent sample of 7 with mean and median both 1. – Nick Cox Aug 16 '21 at 08:07