This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other geometry-discretizing) constraint on the support set. Concretely, we show that a class of such LTV systems is identifiable whenever the upper uniform Beurling density of the delay-Doppler support sets, measured uniformly over the class, is strictly less than 1/2. The proof of this result reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Moreover, we show that this density condition is also necessary for classes of systems invariant under time-frequency shifts and closed under a natural topology on the support sets. We furthermore show that identifiability guarantees robust recovery of the delay-Doppler support set, as well as the weights of the individual delay-Doppler shifts, both in the sense of asymptotically vanishing reconstruction error for vanishing measurement error.