There's not really an in-depth or comprehensive answer to give here, because no, there was no specific tradition of clothing used to distinguish English families. I discuss what could be thought of as English national dress in this previous answer, however. But there was also no tradition of "clan tartans" prior to the nineteenth century.

While tartan was an important visual aspect of Highland Scottish culture from even before the Jacobite Rising, the concept of the clan tartan comes from certain publications by James Logan, John Carter Allen, and Charles Manning Allen (the latter two under the infinitely more romantic names "John Sobieski Stuart" and "Charles Edward Stuart") - The Scottish Gael, or Celtic Manners as Preserved among the Highlanders (1831), followed by The Vestiarium Scoticum (1842), The Costume of the Clans (1845), and The Clans of the Scottish Highlands (1845-1847). Even at the time, the assertion in these books that clan tartans were a thing was debunked by other writers, but the notion persisted - most likely due to the extreme romanticization of Scotland once the Rising and the Clearances were long past. Because the clan tartan never existed outside of this context, it's therefore unsurprising that English families never developed anything along the same lines.