Interface-confined superconductivity emerges from the interaction of electronic states across chemically distinct boundaries, providing a route to engineer superconducting phases where magnetism and topology coexist. Determining the intrinsic nature of such superconductivity, however, is challenging because the superconducting layer is only a few nanometers thick and buried beneath several normal layers. Here, we measure the whole-sample Meissner response of six FeTe-based heterostructures in a uniform magnetic field using a frequency-domain tunnel-diode resonator. In the ultrathin limit, the conventional normalization of the measured susceptibility, χ(T→0)=-1, fails by tens of percent. We establish the appropriate calibration and invert χ(T) to determine the London penetration depth λ(T). Two key results emerge. First, the broad transitions observed in χ(T) arise naturally from the extreme geometry and large λ, without requiring chemical or structural inhomogeneity; the extracted λ(T) closely tracks the resistive transition. Second, λ(T) and the corresponding superfluid density are inconsistent with a fully gapped isotropic s-wave state and instead indicate a strongly anisotropic order parameter possibly with line nodes or deep gap minima. The inferred λ(0) is of order 1 μm, consistent with an independent analysis of the 2D phase stiffness. Despite the distinct chemical, magnetic, and topological character of the three overlayers, all six FeTe heterostructures exhibit similar low-temperature power-law behavior, with no systematic dependence of the superconducting response on overlayer identity. These results point to the interfacial FeTe layer as the common origin of superconductivity.