I suppose trails in Britain aren't so long as American ones, so there isn't such a time commitment. You would have to be very fortunate, or maybe retired, to take time off work, family, mortgage etc to do a trail which took you 4 or 5 months, whereas here you are usually talking one or two weeks, so it is easier to think in terms of doing the trail in one go.
That said, some people do break them up into week long sections, especially the longer trails.
I like to do a long distance walk every year, sometimes a recognised trail, occasionally just one I made up. But I also like to have one on the go that I do in chunks of 2 or 3 days when there is time and a window in the weather.
So I suppose the answer is both. We just haven't developed the terminology over here.