Voice anonymization aims to conceal speaker identity and attributes while preserving intelligibility, but current evaluations rely almost exclusively on Equal Error Rate (EER) that obscures whether adversaries can mount high-precision attacks. We argue that privacy should instead be evaluated in the low false-positive rate (FPR) regime, where even a small number of successful identifications constitutes a meaningful breach. To this end, we introduce VoxGuard, a framework grounded in differential privacy and membership inference that formalizes two complementary notions: User Privacy, preventing speaker re-identification, and Attribute Privacy, protecting sensitive traits such as gender and accent. Across synthetic and real datasets, we find that informed adversaries, especially those using fine-tuned models and max-similarity scoring, achieve orders-of-magnitude stronger attacks at low-FPR despite similar EER. For attributes, we show that simple transparent attacks recover gender and accent with near-perfect accuracy even after anonymization. Our results demonstrate that EER substantially underestimates leakage, highlighting the need for low-FPR evaluation, and recommend VoxGuard as a benchmark for evaluating privacy leakage.