[standevbob]

Stan provides both frequentist inference (penalized maximum likelihood with bootstrapped confidence intervals) and Bayesian inference (MCMC sampling or approximate variational) inference.

As currymj says, the differential equations (same for all the linear algebra solvers like eigendecomposition) can be used in defining likelihoods for either Bayesian or frequentist estimation. Same for all of our linear algebra operations and special functions.

Not every model that can be programmed in Stan has a well-defined MLE or proper posterior. Standard hierarchical/multilevel models don't have MLEs, even with standard shrinkage. Bayesian models with improper priors and no data wind up with improper posteriors, etc.

Having said all that, almost all of the use of Stan is for Bayesian inference.