The Catalan numbers you get in types other than A depend on which interpretation of the Catalan numbers you are generalizing. There are at least two possibilities: the number of anti-chains in the root poset (these are sometimes called non-nesting partitions) gives the so called Coxeter-Catalan number; these are well-studied, see e.g. "A uniform bijection between nonnesting and noncrossing partitions" by Armstrong, Stump, and Thomas; the number of Stembridge's "fully commutative" elements (in type A, this is the same as 321-avoiding permutations) of the Weyl group is another generalization which gives different numbers.

The Catalan numbers you get in types other than A depend on which interpretation of the Catalan numbers you are generalizing. There are at least two possibilities: the number of anti-chains in the root poset (these are sometimes called non-nesting partitions) gives the so called Coxeter-Catalan number; these are well-studied, see e.g. "A uniform bijection between nonnesting and noncrossing partitions" by Armstrong, Stump, and Thomas. The number of Stembridge's "fully commutative" elements (in type A, this is the same as 321-avoiding permutations) of the Coxeter group is another generalization which gives different numbers, see "The enumeration of fully commutative elements of coxeter Groups" by Stembridge for explicit generating functions in all types.