Black Watch is not a typical Scottish tartan - ie a cloth associated with a particular clan etc. (Though in fact clan tartans as many people think of them today were only invented in the 19th century. Before that, there were regional tartans - the special designs of local weavers - and 'uniform' tartans, given by rich landowners to poor workers/retainers.) The Black Watch tartan is a special type of uniform tartan. It is camouflage, issued to Scottish soldiers/scouts with knowledge of wild and harsh local moor and mountain conditions who joined local 'watches' (companies) of troops loyal to the Hanoverian government after the failed Jacobite rising of 1715. They had a special responsibility for putting down Jacobite rebels. They may have been wild Highlanders but they did not support the Stuarts (Bonnie Prince Charlie etc). Later, these watch companies were amalgamated with regiments in the British army, and then reorganised again in the 1880s.
So the Black Watch tartan is a government tartan. It is associated with Prince William's Hanoverian ancestors BUT ALSO as Minder very importantly says, with the area close to Dundee. Scottish soldiers belonging to the reformed Black Watch, recruited from the lands around Dundee - literally the 'poor bloody infantry' - fought with enormous bravery during World War I and suffered horrendous losses at Loos, the Somme and elsewhere. Their bravery and fierceness - and their kilts - led the Germans to nickname them 'the ladies from hell'.
I don't know whether all this is why Catherine is wearing Black Watch tartan, but many older people (and maybe some younger, too) from Dundee will recognise its sad history.
Wikipedia is quite detailed and well-referenced: en.wikipedia.org/wiki/Black_Watch