If I was in a car accident and the officer that arrived in the didn’t ask for my name just my date of birth, so how do I go about filing a claim against the other car who hit the vehicle I was in?

You don't "file a claim against" someone else's insurance per se--if you know who it was, you can of course communicate with them and request compensation, which they may pay or may pass on to their insurer, and if they do pass it to their insurer, the insurer may choose to  pay...but 1 the other person's insurer is the other person's insurer--they do not owe you a duty and 2 the other person and their insurer could decline to pay you, such as because they do not feel they are at fault. Only if the other person is found to be at fault might he/she have to pay.
If such a voluntary attempt to get settlement does not, or you do not want to bother going through such attempts, what you do is file a lawsuit against the driver whom feel was at fault and/or against the owner of the other car, if the owner is not the same person as the driver. By suing, you force them to respond they may at that point choose to settle, but if they do not, then you will have your chance to prove your case--that is, to prove in court that the other person was at fault and caused certain injuries or losses to you--and get a court judgment requiring the payment of compensation. For amounts less than the limit for your small claims court, a good option is often to sue in small claims, acting as your own attorney "pro se" so as to save on legal fees.