What a great photo! Well done. It has a downspout, so it is a gutter. And look at the adjoining building. The gutter there has a downspout that connects to the funky midroof gutter.
Here is my guess. The gutter was installed by a person much like myself - aware of the fact that water runs downhill and unconstrained by any gutter customs. Needing to have the gutter slope towards the middle of the house where the downspout is (and probably a sewer access, so unmoveable) the gutter starts high enough on the roof that it was easy to get the gutter to slope down towards the center.
The sewer access is the only one around, which is why the adjoining building needs to drain its own gutter into this one.
You will see that my doppelganger in Buffalo has also installed a charming gutter and downspout just for the cover over the front door. Possibly the cover was catching runoff from the roof where the roof gutter was high up and missed it. There is a little green and white wall around the top edge of the door cover to keep the water from going off the front and landing on the steps and freezing.
I love this guy. Dad, is that you?