We know from the Chinese remainder theorem that if an ideal I in a commutative ring $R$ is the intersection of finite many maximal ideals $M_i$ then the quotient ring $R/I$ is isomorphic to the direct product of $R/M_i$. And one can show that the these maximal ideals are uniquely determined.
But for a group $G$, can a subgroup $H$ be expressed as the intersection of some maximal subgroups of $G$ in different ways, e.g. $H = M_1\cap M_2 = M_3 \cap M_4$ with $\{M_1,M_2\}$ different from $\{M_3,M_4\}$? And what is the key difference here compared with the ring case?
The stated property does not hold for groups.
The Klein $4$-group, $C_2\times C_2$, has three maximal subgroups: $C_2\times\{e\}$, $\{e\}\times C_2$, and the diagonal subgroup. The intersection of any two of them is equal to the trivial group. So the trivial group can be expressed as an intersection of maximal subgroups in three different ways. The same is true in $S_3$, where any two of the three subgroups of order $2$ have trivial intersection.
You can even have disjoint families of maximal subgroups with the same intersection. Modifying the first example a bit, take $C_p\times C_p$. The maximal subgroups correspond to one dimensional subspaces, so they are $\{e\}\times C_p$, and all subgroups of the form $\langle (x,x^i)\rangle$ with $0\leq i\lt p$. There are $p+1$ of them, and any two intersect trivially. So for $p\geq 3$, you can find four pairwise distinct maximal subgroups $M_1$, $M_2$, $M_3$, and $M_4$, any two of which have trivial intersection.
Note that the former example uses normal subgroups, so you cannot save the proposition by using normality as an analogue of “ideals”.
(You can get examples where the intersection is not trivial by taking a group that has a nontrivial quotient isomorphic to the Klein $4$-group; for example, the quaternion group $Q_8$, and the subgroups $\langle i\rangle$, $\langle j\rangle$, and $\langle k\rangle$, any two of which intersect at $\{1,-1\}$; the nonabelian group of order $p^3$ and exponent $p$ has a central quotient isomorphic to $C_p\times C_p$, yielding more examples in the same manner.)
What is different about groups and rings? I’m not sure what it is exactly, but here is one major difference: the only ideals of a direct product of rings, $R_1\times\cdots\times R_n$ are of the form $I_1\times\cdots\times I_n$, with $I_j\triangleleft R_j$. This is not true for subgroups (or even normal subgroups) of direct products of groups. Even for groups with no nontrivial subgroups like $C_p$, there are more maximal subgroups of $C_p\times C_p$ than the coordinate subgroups.