Ideals on $\mathbb N$ and large sets that have small intersection

The denumerable atomless Boolean algebra $A$ can be isomorphically embedded in $\wp(\omega)/\mathsf{fin}$. The identity on $A$ can be extended to a homomorphism from $\wp(\omega)/\mathsf{fin}$ into the completion of $A$. This gives a homomorphic image of $\wp(\omega)/\mathsf{fin}$, hence also of $\wp(\omega)$ itself, in which every disjoint subset is countable. So the kernel of the homomorphism is another ideal of the type indicated.

I'm not sure the following example is different from the ones already given, but the description is different, and so I hope some readers might find it useful. Fix a countable family $\mathcal F$ of non-principal ultrafilters on $\mathbb N$, and let $I$ be the ideal of sets that are in none of the ultrafilters in $\mathcal F$. Then $I$ is not maximal, because if $X\subseteq\mathbb N$ is in some but not all of the ultrafilters in $\mathcal F$ then neither $X$ nor its complement is in $I$. (Almost as easily, $I$ is not the intersection of finitely many maximal ideals.)

To prove that $I$ has the countability property in the question, consider any uncountable family $\mathcal A$ of sets not in $I$, Then each set in $\mathcal A$ is in at least one of the ultrafilters in $\mathcal F$. But $\mathcal F$ is countable and $\mathcal A$ is not, so there must be two (in fact uncountably many) sets $A,B\in\mathcal A$ belonging to the same ultrafilter $U\in\mathcal F$. Then $A\cap B$ is in $U$ and therefore not in $I$.

In topological language: for any closed subset, $F$, of $\beta\mathbb{N}\setminus\mathbb{N}$ the family $I_F=\{A:A^*\cap F=\emptyset\}$ is an ideal. Your property translates into: the closed set $F$ is a ccc space.

The example by Monk can be made in this way too: take a map $s$ from $\beta\mathbb{N}\setminus\mathbb{N}$ onto the Cantor set $C$ and take $F$ such that $s:F\to C$ is irreducible and onto, then $F$ is sparable and hence ccc.

The example by Blass corresponds to the closure of a countable set in $\beta\mathbb{N}\setminus\mathbb{N}$, again a separable subspace.

In general: take any compact ccc space, $X$, of weight at most $\mathfrak{c}$ and embed it into the Tychonoff cube $[0,1]^\mathfrak{c}$. There is a continuous map $\sigma$ from $\beta\mathbb{N}$ onto that cube. Now take $F$ such that $\sigma:F\to X$ is irreducible and onto; then $F$ is ccc. For a non-separable example let $X$ be the Stone space of the measure algebra.