On concrete,yes. On soil or in a bed.absolutly no.

Is it possibly connected to an underground pipe, to take the water away from the house?

Photo doesn't tell the story. I can't tell if the downspout ends above the ground or if it goes underground.

If it stops above ground, IMO, it should be turned so as to direct the flow away from the building. Which is the basic reason for gutters.

The other thing that the photos don't show is if the paving slopes away from the building, how far the paving extends beyond the building, where the nearest permeable surface is and if it is adequately sloped.