First, do not overestimate role of DNA; DNA is a huge molecule that carries genetic information in living organisms, but it is not DNA that hold living
organisms together. All things (water, humans, cups, etc...) are composed of
molecules or atoms, and there are numerous forces of intramolecular and
intermolecular attraction that hold molecules together (see https://library.thinkquest.org/C006669/data/Chem/bonding/inter.html
). Molecules are not just balls that are situated one near other, each molecule or
atom is a complicated quantum structure that interacts in many ways with other
molecules and atoms (a scheme is here: https://www.tutorvista.com/content/physics/physics-iii/solids-and-fluids/intermolecular-interatomic-forces.php
). Thus, forces hold molecules together, and these forces have pure physical
(quantum) nature (see list of forces: https://en.wikipedia.org/wiki/Intermolecular_force
). Furthermore, molecular forces are the same in living organisms and abiotic
objects. In modern physics and chemistry there is no need to add some "kind of
DNA that makes matter what it is", because properties of matter are quite good
described by physical and chemical theories.
Next, life is quite unclear thing because in some cases it is almost impossible to define if the thing is living or not. It depends on the personal
opinion of the explorer. We consider that cup is not a living object (because it
does not have characteristics of live that can be found here: https://en.wikipedia.org/wiki/Life#Biology
), but obviously there are many bacteria that live on its surface.
Finally, theoretically the cup could keep record of all the process of creating, but we do not have equipment that can register this information. You
can read the article https://en.wikipedia.org/wiki/Water_memory
about the most famous research in this area.