I thought entanglement involved sharing a similar (or the same?) wave function.
Nothing could be farther from the truth. I could entangle the polarization of a photon with the position of an electron. Or entangle the energy of an electron with the direction of propagation of a photon.
Its really just superposition. A state with an electron on the left and a photon with a vertical polarization is a fine state. So is a state with the electron on the right and the photon having a horizontal polarization. And since both those states are fine, so is a complex superposition. And that particular superposition is an entanglement of the two options.
And here is the key. You could get something similar to entanglement by just flipping a coin and if it is heads making the state with an electron on the left and a photon with a vertical polarization. And if it is tails making a state with an electron on the right and a photon with a horizontal polarization. But that isn't true entanglement, because a true entanglement really is that superposition so it allows you to measure a circular polarization and you'll get your position in a superposition rather than than just getting the left or right you made.
So an entanglement is similar to just making one option or another. But it is different, it has the full superposition of the two possibilities. And this means interactions have access to both.
And there is another key here. A non entangled state could be written as $\left|\text{left}\right\rangle_e\otimes\left|\updownarrow \right\rangle_\gamma$ and it makes sense to say the electron has a state (of being on the left) and the photon has a state (of vertical polarization). And a non entangled state could be written as $\left|\text{right}\right\rangle_e\otimes\left|\leftrightarrow \right\rangle_\gamma$ and it makes sense to say the electron has a state (of being on the right) and the photon has a state (of horizontal polarization). But for the entangled state
$$\frac{1}{\sqrt 2}\left|\text{left}\right\rangle_e\otimes\left|\updownarrow \right\rangle_\gamma+\frac{1}{\sqrt 2}\left|\text{right}\right\rangle_e\otimes\left|\leftrightarrow \right\rangle_\gamma
$$ it doesn't make sense to say the electron has a state or that the photon has a state. Instead the system has a state and that's all we can really say.
In effect, any action on one is no different than an action on the other as if they were the same exact particle at the same location, right?
This is 100% wrong. From the example above you can see that a massive charged electron with spin 1/2 can have its position be entangled with the polarization of a massless uncharged spin 1 particle. That's as different as things can be. The point is that it is similar to just making $\left|\text{left}\right\rangle_e\otimes\left|\updownarrow \right\rangle_\gamma$ or making $\left|\text{right}\right\rangle_e\otimes\left|\leftrightarrow \right\rangle_\gamma$. Similar in that you might get one option or you might get the other and this isn't going to send information from one to the other. If you measure the polarization of your photon and find it is vertical then you could know which pair exists now, but you didn't control which you got, you just learned which you got.
But it isn't as simple as just making one or the other, since you might measure something other than horizontal/vertical polarization. So the entire joint state exists.
What you really have is a joint state that can split into different possibilities depending on how you interact with it. And the fact that it is nonlocal means that an interaction in one region can split the entire state into parts.
But this is because it is a joint state and the individual particles don't have their own states to break into parts. Normally things break into parts but in an entangled system the joint system has to be what breaks into parts.
the entangled particles cannot be separated faster than c.
Now this is a problem too. Firstly, entanglement can be swapped so the particles could become entangled through a swapping of entanglement and the newly entangled particles could be ones that have never ever seen each other. Secondly, a particle might not have a well defined position. If one particle is in superposition of states with very different positions then what does it mean to talk about how far in space it is from something else?