The current privacy evaluation for speaker anonymization often overestimates privacy when a same-gender target selection algorithm (TSA) is used, although this TSA leaks the speaker's gender and should hence be more vulnerable. We hypothesize that this occurs because the evaluation does not account for the fact that anonymized speech contains information from both the source and target speakers. To address this, we propose to add a target classifier that measures the influence of target speaker information in the evaluation, which can also be removed with adversarial learning. Experiments demonstrate that this approach is effective for multiple anonymizers, particularly when using a same-gender TSA, leading to a more reliable assessment.