In this paper, we present an exact (i. e. non-approximated) and linear measurement model for hybrid AC/DC microgrids for recursive state estimation (SE). More specifically, an exact linear model of a voltage source converter (VSC) is proposed. It relies on the complex VSC modulation index to relate the quantities at the converters DC side to the phasors at the AC side. The VSC model is derived from a transformer-like representation and accounts for the VSC conduction and switching losses. In the case of three-phase unbalanced grids, the measurement model is extended using the symmetrical component decomposition where each sequence individually affects the DC quantities. Synchronized measurements are provided by phasor measurement units and DC measurement units in the DC system. To make the SE more resilient to vive step changes in the grid states, an adaptive Kalman Filter that uses an approximation of the prediction-error covariance estimation method is proposed. This approximation reduces the computational speed significantly with only a limited reduction in the SE performance. The hybrid SE is validated in an EMTP-RV time-domain simulation of the CIGRE AC benchmark micro-grid that is connected to a DC grid using 4 VSCs. Bad data detection and identification using the largest normalised residual is assessed with respect to such a system. Furthermore, the proposed method is compared with a non-linear weighted least squares SE in terms of accuracy and computational time.