This is the legal name for chocolate which contains a higher milk content (up to 20%) than Continental recipies. Family milk chocolate is the standard chocolate in the UK and in Ireland, made by manufacturers such as Cadbury. It can still be sold as milk chocolate in its home countries, but must use the additional term "family" when sold on the Continent.
In addition to the extra milk, the manufacturers of family milk chocolate add up to 5% vegetable oils (other than cocoa butter). Chocolate with this level of vegetable oil must be labelled as such. The words "contains vegetable fats in addition to cocoa butter" must appear in the same field of view as the ingredient list and product name.
This terminology and labelling changes ended a dispute about chocolate sales that ran from 1973 until 2000.
Source: MAFF press release, March 15, 2000