Unfortunately, classical analogies generally can't capture the interesting properties of quantum systems and these analogies are no exception. One challenge is that quantum states can look like they obey classical randomness if you don't follow the right steps. This means that classical analogies for quantum mechanics sometimes right, but never in interesting ways.
To be more specific, here's where things go wrong (you will need familiarity with density operators):
Superposition Classical randomness: A coin flipped in the air can be modelled by an unknown value (a "random variable") $X$, which can be treated as either heads or tails until it is observed. Saying that $X$ can be both heads and tails "at the same time" is not very well defined, but we can reason about random variables in a way that accounts for all possible values they might take. And a convenient way for doing so is to use density operator notation from quantum mechanics,
$$
\rho_X := \begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \end{pmatrix}. \tag{1}
$$
This object can be "measured" in exactly the same way as a quantum state is, and until it is measured it exists as a mixture of $|0\rangle$ and $|1\rangle$. But this object obeys all of the laws of QM and can be treated as a quantum system(!), just not a very interesting one. Compare this to the density matrix for the state $|+\rangle = (|0\rangle +|1\rangle)/\sqrt{2}$,
$$
|+\rangle \langle +| = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}. \tag{2}
$$
This state exists as a superposition of $|0\rangle$ and $|1\rangle$, which is closely tied to those off-diagonal elements. The problem is, if all you do is measure $\{|0\rangle, |1\rangle\}$ (i.e. look at the result of a coinflip), you won't see a difference between (1) and (2), so the analogy has managed to trick us in this case.
Entanglement Classical correlation: Just like we modelled a coinflip with a random variable $X$, we can model classical correlations between two coins that have the property that you mentioned. Simply flip an unbiased coin $X_1$, give it to someone else (so that you don't see the result) to prepare a coin with the same outcome, $X_2 = X_1$. Then send $X_2$ as far away as you like. Looking at either coin immediately reveals the result of the other. As before, this scenario has a density matrix representation,
$$
\rho_{X_1 X_2} = \begin{pmatrix}
1/2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1/2
\end{pmatrix}. \tag{3}
$$
So this system exists as a classically correlated mixture of $|00\rangle$ and $|11\rangle$ until a measurement is performed. We can compare this to a genuinely entangled state $|\Phi\rangle = (|00\rangle +|11\rangle)/\sqrt{2}$ represented as
$$
|\Phi\rangle \langle \Phi| = \begin{pmatrix}
1/2 & 0 & 0 & 1/2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1/2 & 0 & 0 & 1/2
\end{pmatrix}. \tag{4}
$$
This state is entangled, which is closely tied to the off-diagonal elements again. But if all you ever do is check for heads or tails (i.e. perform a measurement in a single basis), these states (3) and (4) behave in the same way, and so the analogy has again tricked us. But there isn't a widely accepted classical analogy that fully captures the interesting properties of entanglement, which require e.g. using different bases for measurement.
Constructing analogies like this might be more pleasant than dealing with the unfamiliarity and un-intuitiveness of quantum mechanics, but classical analogies cannot really convey non-classical concepts.
