I see this in nanny job adverts but it's usually the agency using those terms- based on speaking with the family and wanting to make the best match. In general I've found families described as 'formal' tend to be those where there are few signs of the family having children- I've seen houses with living rooms which the children are clearly not allowed in, one family that you had to carry the buggy upstairs every time you brought it in so it was out of sight! They tend to be the ones looking for 24 hour care, as in the nanny gets the monitor at night and gets up for the kids.
The relationship tends to be different too, the more formal houses seem to be the ones that require you to ask permission for play dates, or to take the children to new places. They also seem to have more staff, nanny plus housekeeper plus gardener, in one case 2 nannies, gardener, full-time housekeeper, accounts assistant once a week plus a cleaner! For a house with one child! If I were to describe the family to the next nanny or something I would definitely say they were a formal family.
My preference is to find those that don't say formal or informal, just a family! Seems to be by area too, I interviewed for several families I would describe as formal in the Kensington/Knightsbridge kind of areas.