Speaker anonymization aims to protect speaker identity while preserving content information and the intelligibility of speech. However, most speaker anonymization systems (SASs) are developed and evaluated using only English, resulting in degraded utility for other languages. This paper investigates language mismatch in SASs for Japanese and Mandarin speech. First, we fine-tune a self-supervised learning (SSL)-based content encoder with Japanese speech to verify effective language adaptation. Then, we propose fine-tuning a multilingual SSL model with Japanese speech and evaluating the SAS in Japanese and Mandarin. Downstream experiments show that fine-tuning an English-only SSL model with the target language enhances intelligibility while maintaining privacy and that multilingual SSL further extends SASs' utility across different languages. These findings highlight the importance of language adaptation and multilingual pre-training of SSLs for robust multilingual speaker anonymization.