I think this is likely a cloth meant to be a headband (hachimaki) because of its size, and the fold across the middle. But it could just be a hand towel as well. A sennninbari would likely have embroidery on it. But it is a minor point. All of these items are meant to be good luck charms for the wearer.
Regarding Natsugumo, I think its just a coincidence that it is also the name of a ship. This is because the red seal on your cloth also says "Natsugumo" and this red seal is usually the personal seal of the artist. Also, the ship and its administration wouldn't be presenting any sailor with a cloth for good luck. The cloth is presented by the family.