Why do galaxies and water going down a plug hole spin?

We all experience things spinning, whether it's water down a drain, the earth on its axis, planets round the sun, or stars in a galaxy - even electrons round an atom.

But why is spin so common in the universe?

I think there are some physical laws governing this, but perhaps my question is deeper than stating and explaining the law (conservation of angular momentum).

an excellent question.

First, let me begin with the electron because quantum mechanics is what I've been trained to do. If you find this portion inpenetrable, jump to the section Classical physics.

Electrons have an internal spin equal to 1/2. But when it comes to their rotation around the nucleus, electrons actually don't spin around the hydrogen nucleus or helium nucleus at all. The ground state has electrons in the "1s" state and the letter "s" means that the orbital angular momentum is zero.

However, it's true that "most states" - like excited states of the hydrogen, and most states of other atoms - have spinning electrons. The reason is not hard to be explained mathematically: the angular momentum is a function of the other basic quantities ("observables") that may change. And because the spin may be nonzero, it usually is nonzero. One needs to fine-tune things very accurately for the spin to be exactly zero and such things don't occur naturally.

The angular momentum in quantum mechanics is quantized and may be described by two quantum numbers, $l, m$, where $m$ goes between $-l$ and $+l$ with the spacing equal to one. Here, $m$ says what is the $z$-component of the angular momentum and $l$ determines its total magnitude. (The components relatively to other axes may be measured as well - $z$ is just a convention - but they cannot be measured simultaneously because the operators don't commute i.e. because of the uncertainty principle.)

The states with the angular momentum given by $l,m$ are described by wave functions that depend on the angular variables by the function $Y_{lm}(\theta,\phi)$, the spherical harmonics. Whenever $l,m$ is different from $0,0$, i.e. whenever the angular momentum is nonzero, the function $Y$ is non-constant, and vice versa. If you have anything that depends on the direction in quantum mechanics, it really implies that the angular momentum is nonzero.

Classical physics

The angular momentum is conserved - both in classical physics and quantum physics - and the deeper reason for that is Noether's theorem. Emmy Noether has demonstrated that all systems whose laws are rotationally symmetric will exhibit a conserved quantity called the angular momentum, and vice versa. (She has done the same thing with many more - as well as the most general - pairs of symmetries and conservation laws, too.)

I am convinced that the rotational symmetry of the laws of physics is a deep thing - phenomena don't depend on the direction.

Your universalistic summary of many different "spinning systems" surely sounds intriguing but I think that it is really not so deep from a physics viewpoint. The deep thing is that the angular momentum is conserved and why; but why a particular system has a high chance to end up spinning depends on the system and its details.

In particular, both the galaxies and the whirlpool are spinning for a very similar reason, indeed. It is the same reason why your spinning rate increases while you're skating and you shrink all your arms and hands closer to the axis of the rotation. Why? Well, the angular momentum is conserved. But it may be written as $J=I\omega$ where $J$ is the angular momentum, $I$ is the moment of inertia, and $\omega$ is the angular frequency.

If you shrink your arms and legs, you actually reduce $I$ because $I\approx MR^2$ where $M$ is the mass and $R$ is the rough distance of the mass from the axis. If you shrink $R$, by getting closer to the axis, you will also reduce $I$, and to keep $J$ constant which is a law of physics, you have to increase $\omega$: you will be spinning faster.

The same thing occurs with water in the bath tub. A liter of water that is approaching the sink was pretty much everywhere and had a reasonable angular momentum because of the chaotic motion of the water. However, when you suddenly compress all of the water in the sink - very near the axis of the sink - the liter of water that is just leaving will reduce its $R$, distance from the axis, and also $I$ as a consequence. So the angular frequency $\omega$ inevitably increases.

Similarly, for the galaxies that are being formed, lots of gas is moving in random directions, but the gravitational collapse shrinks the matter so that it is closer to the axis of the future galaxy. Because it is closer, the moment of inertia decreases, so the angular frequency has to increase just like before.

So indeed, there is a similarity - and the same explanation - for many phenomena where things begin to spin more quickly than before. And these phenomena are cute and intriguing. But physics may explain them and the truly "beautiful" and "deep" explanations are much more abstract in character.