The generalized linear system (GLS) has been widely used in wireless communications to evaluate the effect of nonlinear preprocessing on receiver performance. Generalized approximation message passing (AMP) is a state-of-the-art algorithm for the signal recovery of GLS, but it was limited to measurement matrices with independent and identically distributed (IID) elements. To relax this restriction, generalized orthogonal/vector AMP (GOAMP/GVAMP) for unitarily-invariant measurement matrices was established, which has been proven to be replica Bayes optimal in uncoded GLS. However, the information-theoretic limit of GOAMP/GVAMP is still an open challenge for arbitrary input distributions due to its complex state evolution (SE). To address this issue, in this paper, we provide the achievable rate analysis of GOAMP/GVAMP in GLS, establishing its information-theoretic limit (i.e., maximum achievable rate). Specifically, we transform the fully-unfolded state evolution (SE) of GOAMP/GVAMP into an equivalent single-input single-output variational SE (VSE). Using the VSE and the mutual information and minimum mean-square error (I-MMSE) lemma, the achievable rate of GOAMP/GVAMP is derived. Moreover, the optimal coding principle for maximizing the achievable rate is proposed, based on which a kind of low-density parity-check (LDPC) code is designed. Numerical results verify the achievable rate advantages of GOAMP/GVAMP over the conventional maximum ratio combining (MRC) receiver based on the linearized model and the BER performance gains of the optimized LDPC codes (0.8~2.8 dB) compared to the existing methods.