Are we sure that this brooch/badge is related to the Duchy of Lancaster?
First, this brooch, though it bears an heraldic lion, does not look like the coat-of-arms of the Duchy of Lancaster (left), nor of the County council of Lancashire (right):
Besides, Queen Elizabeth II wore this brooch not only during visits to Lancashire (in 2006 and 2015, see the pics posted by Joye and Beth), but also during a visit to Scarborough, in North Yorkshire (so, not in Lancashire...) in May 2010.
A remarkable detail is that, during this visit to Scarborough, North Yorkshire, in May 2010, the Queen swapped this badge with the Jardine Star Brooch (but I do not know at which stage of the visit, and which of the two brooches was worn first):
According to the captions of the pictures on Getty's website, the Queen wore the "Lancaster" badge at the Cloughton blacksmiths shop, whereas she wore the Jardine Star Brooch for the reinauguration of the renovated open-air theatre of Northstead Manor Gardens. There must probably have been a specific reason for this very unusual brooch swap, but I do not get which one...