GMT Project projecting all data points to cross-section

What do you mean by “not correct”. Since your blue blob starts around lat = 47 and ends at lat = 63, it will fatten up by cos(47)/cos(63) ~ 1.5.

what I meant is that the dashed black rectangle appears to widen as you move towards the north, whereas my expectation was to achieve a rectangle similar to the cyan one. I guess it is just a matter of projection then?

Probably. It will certainly widen as we plot northward on a Mercator projection

I think you are after buffer polygons. See gmtspatial -S. Not sure if there is an option to trim the half-circle tips.