Does your father have citizenship? You need to have at least one parent with the ID card to be able to apply for (dual) citizenship and there’s no need to give up your current mysterious citizenship. If you don’t have a parent then you will have to go the naturalization route giving up your current citizenship but that process is completely different. You would not get a Taiwanese passport first and you have to live in Taiwan for something like 5 or 6 years before you can become a citizen.
Read this thread.
[url]How to get Taiwan Citizenship - Primer, FAQ, and Resources
Applying as an overseas Taiwanese seems to me a much easier route.