'Britishness' - like all national concepts - is inherently indefinable. It's reliant on generalisations like the 'stiff upper lip' ones you've described in the title post, which have only ever been true to an extent.
It's also defined in different grounds. Ask someone on the right what 'Britishness' is or was and they might talk about national pride or how great the country was/is. Ask someone on the left and they might cite the UK's record on civil liberties and social legislation.
Less politically, others might also cite national symbols (the monarchy, historic figures etc). All of these seem rather superficial, though.
On the other hand, I think most people would agree there's a British 'culture' - or perhaps more likely that there are English, Irish, Scottish and Welsh cultures - and that these are also pretty difficult to define yet we do tend to agree they do exist. If this is the case, then you might say 'Britishness' is similar.