This property of CABAs is not applicable in constructive mathematics, where power sets are rarely boolean algebras. However, we can use discretelocales instead (or rather, their corresponding frames). That is, define a CABA to be (not a complete atomic boolean algebra but) a frame $X$ such that the locale maps $X \to 1$ and $X \to X \times X$ (which in the category of frames are maps $0 \to X$ and $X + X \to X$) are open (as locale maps). Then it should be (I will check) a classical theorem that CABAs and complete atomic boolean algebras are the same, and a constructive theorem that CABAs and power sets are the same (in the same functorial manner as above).

Algebraicity

Complete Boolean algebras are the models of an algebraic theory (in which the operations, notably $j$-indexed suprema and infima, have arities $j$ unbounded by any cardinal). It follows from general principles that the underlying-set functor $U: CompBoolAlg \to Set$ preserves and reflects limits and isomorphisms.

However, this functor $U$ is notmonadic; in fact, it does not even possess a left adjoint. Indeed, while the free complete Boolean algebra on a finite set $X$ exists and coincides with the free Boolean algebra on $X$ (it is finite, being isomorphic to the double power set $P(P X)$), we have

Theorem (Gaifman-Hales; Solovay)

There is no free complete Boolean algebra on countably many generators.

As a consequence, $CompBoolAlg$ is not cocomplete (otherwise there would exist a countable coproduct of copies of $P(P 1)$, which is ruled out by the previous theorem).