Entanglement is the same thing.
Not quite. I think it is best seen by the Mermin paradox:
Three particles are brought into a special shared quantum state (termed GHZ state) and then distributed to three parties, who each can then make, on their own choice, one of two measurements on the particles, X or Y. Either measurement can result either in the value 1, or the value -1.
Now it turns out that while the individual results are completely random, whenever any two of them choose the measurement Y and the third one chooses X, the product of all three measures values are 1, every time.
Now, so far there's no problem: This could easily be explained by the original procedure producing not really the same state, but randomly different states which determine all measurement results, and which all fulfil the condition. This would be the analogue to your coin: Every actual state (heads up or tails up in the case of the coin, the set of six potential measurement results in the case of the Mermin paradox) fixes every measurement result, and all states fulfil a certain condition (the opposite sides of the coin having different symbols, the products of XYY-type measurements being 1 for the Mermin paradox), but the states are otherwise chosen by random. Due to the restriction on the states, you can predict one measurement result if you know the other(s) (for the coin, the down-facing symbol if you know the up-facing, for the Mermin paradox the third measured value of an XYY-type measurement if you know the other two).
Assuming this explanation, let's figure out what the product of measurement results should be if all three people measure X. To this end, let's label as x1 the measurement result the first person got from measuring X, y1 the result the first person would have gotten if measuring Y (which, in the above scenario, would be well-defined, just as in the case of the coin the symbol facing up is well defined even if you don't look at it), x2 the second person's result from measuring X, and so on.
Now we already know that y1*y2*x3=1, y1*x2*y3=1 and x1*y2*y3=1. If we multiply those three values together, we get x1*y1^2*x2*y2^2*x3*y3^2=1. But since the measurement results are all either 1 or -1, their squares are always 1, and thus we end up with x1*x2*x3=1. So according the above explanation, when all three people measure X, the product of their measurement results should be 1, every single time.
Now for the specific quantum state quantum mechanics predicts something different (and experiments confirm it, of course only within measurement error): When all three people measure X, the product of their measurement results is -1, every single time.