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If an oxygen atom has six electrons, then it has an unfilled orbital and the oxygen atom may share electrons from two hydrogen atoms (and form water) in order to become more stable.

But why does oxygen, or any other atom for that manner, favor stability over instability? Why can't the oxygen atom just have six electrons and be done with it?

Also, is an atom being stable identical to it being neutral?

An $text{O}^{2+}$ ion, with 6 electrons, is stable in a vacuum. If you put one in the depths of interstellar space, far from any other matter, then it will sit in that state until the end of time.

Similarly, you can have a high-elevation lake, and it can be absolutely stable if there is no path for water to drain out.

Also, is an atom being stable identical to it being neutral?

So no, as shown by that example, it is not the same thing.

However, of you have a higher-elevation lake and another one whose water level is 10 meters lower, and they're connected by a stream bed, then water will drain out of the higher one until the two surfaces reach an equal level. The stable state is the one that has the minimum energy (minimum total gravitational potential energy of all the water). This is analogous to what happens when oxygen and hydrogen are mixed. The electrons are like the water.

Wanting or favoring are not very good terms in physics. More scientific view on this would be that whenever an oxygen atom comes close to another atom, they interact and provided things are right (such as the number of electrons and their state), the atoms attract and can get closer, while losing part of their initial energy, in the form of radiation or lost electron or transfer some energy to other atom/molecule in a scattering event.

After the bonding energy is lost, it becomes more probable the system will stay together, until required energy for its breaking is supplied from outside. For the air in troposphere, the available mechanisms to supply so much energy (collisions, cosmic particles) can only do so for very little fraction of molecules, so majority of oxygen atoms will exist in pairs, much lower number in triplets and so on.

The oxygen atoms when get close to each other, would certainly interact. When they interact, their is a higher probability that the electrons are shared between the atoms. When they share, they will lose energy.
They will not break apart unless energy comes again from somewhere and satisfies them to stay separated.

And no being stable is not equal to being neutral. Take oxygen atom for example, it is more stable in the ion form than the neutral oxygen atom.

Also, is an atom being stable identical to it being neutral?

No, not at all: many atoms are more stable after they've either absorbed (an) electron(s) or shed (an) electron(s).

A good example is the formation of table salt, $mathrm{NaCl}$, aka sodium chloride.

This compound forms when sodium atoms lose an electron (the valence electron), as in:

$$mathrm{Na}to mathrm{Na^+}+ mathrm{e^-}$$

Similarly the choride atoms can absorb an electron:

$$mathrm{Cl_2}+ 2mathrm{e^-}to 2mathrm{Cl^-}+ 2mathrm{e^-}$$

When those ions combine we get:

$$2mathrm{Na}+mathrm{Cl_2}to 2mathrm{NaCl}+Delta H$$

$Delta H$ is the energy released in the process. The arrangement of these elements in the ionic lattice $mathrm{NaCl}$ is more stable than the combination of the (unreacted) elements.

By losing its 'lone' $mathrm{3s^1}$ valence electron, sodium takes on the very stable electron configuration of neon. Similarly, by absorbing an electron into its $mathrm{3s^23p^5}$ valence electrons, it assumes the very stable electron configuration of argon.