What did he think he was signing at the police station? (He must have thought something, even naive 20-year-olds don't go around signing documents at police stations without having at least a vague idea - no matter how erroneous - of what those documents are about.) And obviously the name of the woman he was accidentally marrying must have been all over those documents - why did he think that was?
And why was this document-signing ceremony such an "ordeal" (your word)? If at the time your husband thought it was just some nothing thing, why is it now an ordeal?
I'm not saying I don't believe your husband's story, but these are questions the Philippine officials are going to ask your husband. And they are going to want good, solid, evidence-based and believable answers from your husband before they make any decisions regarding whether or not to grant him an annulment and/or put him in jail. (And really, I think you might want to know these answers yourself before he goes there. Said answers might affect your decision about whether or not to re-marry him.)
And regarding the jail thing? I don't know about the US but in the Philippines it's a good possibility, especially if the Philippine authorities find out your husband married again without first divorcing his Filipina wife. Marriage contracts are taken very very seriously in the Philippines - for example, someone who commits adultery, even just once, can be imprisoned as a criminal for up to 20 years. So given this fact, I can just imagine what they might do to someone who abandons his Filipina wife to go off to America to marry someone else (which is how the Philippine authorities will see it).
(And yes I know, you're gonna say it's the wife who wants the annulment so it should be all on her, but the authorities there are gonna see annulment and criminal breach of marriage contract as two separate issues - they may well grant the annulment and send your husband to prison.)
Your husband needs to consult with an attorney well-versed in Philippine marriage law. If he actually goes to the Philippines (which is what he'll have to do to have any hope of getting an annulment), he will need to retain the services of a lawyer with a license to practice in the Philippines. And meanwhile, you need to try and get answers to the questions I noted above in my first and second paragraphs.