For text enrollment-based open-vocabulary keyword spotting (KWS), acoustic and text embeddings are typically compared at either the phoneme or utterance level. To facilitate this, we optimize acoustic and text encoders using deep metric learning (DML), enabling direct comparison of multi-modal embeddings in a shared embedding space. However, the inherent heterogeneity between audio and text modalities presents a significant challenge. To address this, we propose Modality Adversarial Learning (MAL), which reduces the domain gap in heterogeneous modality representations. Specifically, we train a modality classifier adversarially to encourage both encoders to generate modality-invariant embeddings. Additionally, we apply DML to achieve phoneme-level alignment between audio and text, and conduct extensive comparisons across various DML objectives. Experiments on the Wall Street Journal (WSJ) and LibriPhrase datasets demonstrate the effectiveness of the proposed approach.