How common it is overall I can't easily say, but applications are easy to mention: e.g. median income as calculated from incomes and weights. 

Let's keep an example calculation trivial: 

Income bins 1, 2, 3, 4 with weights 0.3, 0.4, 0.2, 0.1. 

So cumulative weights are 0.3, 0.7, 0.9 and 1 and cumulative weight 0.5, which defines the position of the median, falls within bin 2, which is reported as the median. 

Interpolation can be as fancy as you like, e.g. linear interpolation between bin limits, and so on. 

In short, order the values, and scale weights to sum to 1. Then cumulative weight 0.5 defines the position of the median, the precise recipe depending on local tradition and personal taste. 

The recipe can be extended to weighted quantiles. 

Various common areas: 

Data are released for geographic areas. We might even try to get a weighted median from area means and their weights. 

Data are released only binned in some other way, e.g. to respect confidentiality, etc. 

Incomes are top-coded, so there are no precise details released on the very richest people. In this case, means are out of the question except with strong assumptions, but medians might be tractable (and interesting and useful any way).