If $G$ is simply connected semisimple and admits holomorphic discrete series, then the tensor product of the holomorphic discrete series of two weights decomposes as a direct sum of other holomorphic discrete series, each appearing with finitely mulitplicity. (In short, holomorphic discrete series comport themselves in general just as in the case of $SL_2(\mathbb R)$.)
I forget the reference now; sorry!
I think that for other Harish-Chandra modules, tensor products have infinite multiplicities (or in more analytic terms, involve direct integrals rather than direct sums). Even the generic principal series for $Sp_4(\mathbb R)$ should demonstrate this behaviour, if I remember correctly.
