Is it possible to use Bayesian Optimization as a generic solver for traditional constrained optimization problems like LP, QP, MIP, etc...or it is it limited in scope to Hyperparameter search and auto-ML problems?
I believe that the OP is referring to the optimization algorithm that builds a Gaussian Process (GP) surrogate model to a black box function and then optimizes the surrogate model. This approach has been extended from unconstrained optimization to include constrained problems as well.
Bayesian Optimization has the advantage that it can be used on an arbitrary function $f(x)$ that can only be computed in black-box fashion. It has the disadvantage that it fails to make any use of known structure in the objective function or constraints.
You can certainly apply this kind of Bayesian Optimization to a more highly structured problem, but chances are that specialized algorithms will perform better than Bayesian Optimization for your problems because these specialized algorithms can take advantage of the structure of the problem.
Bayesian optimization generally refers to the optimization problem that's derived when you apply Bayesian theory to your statistical problem (e.g., some machine learning problem). When you formulate the problem using Bayesian theory, the problem you end up with having can be LP, QP, or MIP, etc.