Reduced Order Models (ROMs) form essential assets across engineering domains, facilitating computation as surrogates of computationally intensive simulators. While the process of reduction can be purely data-driven, physics-based approaches can be leveraged for imprinting physical connotations on the ROM. One such path is offered via projection-based reduction (pROMs), which is shown to achieve a precise approximation of parameterized computational models. However, reduced basis (RB) methodologies can become inefficient or fail in the case of multi-parametric dependencies, or strongly nonlinear systems, while typically lacking a quantification of confidence in the delivered estimate. Projection-based schemes rely on a library of local reduction bases produced via a singular value decomposition on generated Full Order Model (FOM) simulations (snapshot) across samples of the parameter space. The performance of such ROMs largely depends on the technique utilized to relate the parameter vector to the suitable local subspace, with clustering and interpolation strategies adopted in consequence. In this work, we propose the use of Conditional Variational Autoencoders (CVAEs) to achieve a mapping that delivers a continuous mapping on the parameter vector, while comprising a probabilistic nature. In addition, the proposed CVAE-based pROM employs a parametric space that is linked to features that can be measured from actual monitored systems. In this sense, a generative reduced order model is delivered, which retains the physical insights of a projection-based strategy, while readily being coupled to monitoring strategies and sensing-based features. To recover the link to the physical model parameters, an auxiliary task is introduced that uses feed-forward neural-network-based parametrization of suitable probabilistic distributions. These components lead to an efficient and generalized ROM representation with high utility for Structural Health Monitoring (SHM) tasks. The derived ROM is validated in case studies featuring damage in the form of stiffness reduction or exhibiting plasticity/hysteresis under multi-parametric dependencies relating to the system's configuration (geometry, materials) and the characteristics of the input load.