Spoofing detection systems are typically trained using diverse recordings from multiple speakers, often assuming that the resulting embeddings are independent of speaker identity. However, this assumption remains unverified. In this paper, we investigate the impact of speaker information on spoofing detection systems. We propose two approaches within our Speaker-Invariant Multi-Task framework, one that models speaker identity within the embeddings and another that removes it. SInMT integrates multi-task learning for joint speaker recognition and spoofing detection, incorporating a gradient reversal layer. Evaluated using four datasets, our speaker-invariant model reduces the average equal error rate by 17% compared to the baseline, with up to 48% reduction for the most challenging attacks (e.g., A11).