Water itself is not wet. It causes other things to be wet.

Nah, water is neither wet nor dry. First of all, it is not solid to be wet but it is created by moisture. Second, water on very dry surface can't make it wet but there is moisture. Lastly, water cycle.

Yes, water is wet.

First, consider that what we mean by wet is that water (in its liquid state) both coheres to itself, and adheres to things, including being absorbed into them.

Polarity

This is because water is a polar molecule. The water molecule is made of two hydrogen atoms and one oxygen atom joined by covalent bonds. Because the bonds are covalent, the end of the molecule with two hydrogen atoms is slightly positive,(+2) and the end with the oxygen atom is slightly negative (-2).

This causes both The positive end of one molecule adheres to the negative ends of other molecules.

Hydrogen Bonds

Because hydrogen molecules have only one electron, and the innermost shell "holds" a maximum of two electrons, the charge each hydrogen has is +1, which makes it weakly attracted to anything with a negative charge.

Cohesion

Water coheres because the positive ends of water molecules are attracted to the negative ends. Because the weak hydrogen bonds are easily broken, the water molecules stick to each other, but then separate if only a little energy is applied, and then they stick to other water molecules nearby.

Importantly, because the oxygen end has a charge of -2, so it can form hydrogen bonds with hydrogen ends (+1 each) of two different water molecules at a time. This enables water to make a constantly shifting network.

This is what makes water liquid.

Adhesion

The polarity of the water molecule is what causes it to stick to other substances. One or the other charged ends sticks to any molecule that has the opposite charge.

Wetness

Wetness is the quality of being covered by or infused with a liquid (e.g., water).

Because water molecules cohere to each other, they make each other wet, both as individuals and in aggregate.

The combination of adhesion and cohesion of water is how water soaks into things, covers things and thus makes them wet.

Once one water molecule adheres to the substance, other water molecules that cohere to that molecule are pulled by the attraction of the water molecules' polarity into/onto the substance, where they also stick to the molecules of the substance.

The cohesion of water causes more water molecules to stick to the substance than actually touch the substance. This makes things wet.

The exceptions to this are substances made of uncharged molecules. Water cannot make these things wet. Because the molecules are uncharged, the water molecules are not attracted to them, and therefore do not stick. That (combined with cohesion) is why water beads up on these substances.